The sum of absolute maximum and absolute minimum values of the function f(x) = |2x 2 + 3x + 2| + sin x cos x in the interval [0, 1] is :

3 + 1 2 ( 1 + 2 c o s ( 1 ) ) s i n ( 1 )

3 + s i n ( 1 ) c o s 2 ( 1 / 2 ) 2

5 + 1 2 ( s i n ( 1 ) + s i n ( 2 ) )

2 + s i n ( 1 2 ) c o s ( 1 2 )

The sum of absolute maximum and absolute minimum values of the function f(x) = |2x 2 + 3x + 2| + sin x cos x in the interval [0, 1] is :

3 + 1 2 ( 1 + 2 c o s ( 1 ) ) s i n ( 1 )

3 + s i n ( 1 ) c o s 2 ( 1 / 2 ) 2

5 + 1 2 ( s i n ( 1 ) + s i n ( 2 ) )

2 + s i n ( 1 2 ) c o s ( 1 2 )

Class 12 Maths Applications of Derivatives – Step-by-Step NCERT Solutions

Q: 27. Prove that y=4sinθ(2+cosθ)−θ is an increasing function of θ (0,π2) .

We have, y = 4sinθ(2+cosθ)−θ Differentiating w rt.Ø we get, dydθ=ddθ[4sinθ2+cosθ−θ] =(2+corθ)ddθ(4sinθ)−4sinθddθ(2+corθ)(2+cosθ)2−dθdθ. =(2+cosθ)(4cosθ)−4sinθ(−sinθ)(2+cosθ)2−1 =8cosθ+4cos2θ+4sin2θ⋅(2+cosθ)2.−1 =8cosθ+4(cos2θ+sin2θ)−(2+cosθ)2(2+cosθ)2 =8corθ+4−[4+cos2θ+4cosθ](2+corθ)2. =8cosθ+4−4−cos2θ−4cosθ(2+cosθ)2 =4cosθ−cos2θ(2+cosθ)2=cosθ(4−cosθ)(2+cosθ)2 When θ∈[0,π2] we know that, 0≤cosθ≤1. So, 4−cosθ>0 And also, (2 + cos θ)2> 0. ∴dydθ≥0 ∀θ∈[0,π2] Hence, y is an increasing fxn of θ in [0,π2]

Q: 74.Find the maximum and minimum values, if any, of the following functions given by. (i) f(x)=(2x–1)2+3

(i) We have, f(x) = (2x - 1)2 + 3. For all x∈?, (2x−1)2≥0 (2x - 1)2 + 3 ≥ 3. f(x) ≥ 3. ∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = 12 Again as x→∞,f(x)→∞ as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values. (ii) f(x)=9x2+12x+2 A(ii) We have, f(x) = 92 + 12x + 2. f(x)=9[x2+12x9+29] (Taking 9 common from each team). f(x)=9[x2+4x3+29] f(x)=9[x2+2×2x3+(23)2−(23)2+29] f(x)=9[(x+23)2−49+29] f(x)=9[⋅(x+23)2−29]=9(x+23)2−2 For all x∈?,(x+23)2≥0 ⇒(x+23)2−2≥⋅ 0−2 ⇒ f(x)≥ - 2. ∴The minimum value of f(x) = -2 when x+23=0 And as x→∞,f(x)→∞ so f(x) has ⇒x=−23. no maximum values. (iii) f(x) = (x - 1)2 + 10 A(iii) we have, f(x) = - (x - 1)2 + 10 For all x∈?, (x−1)2≥0. ⇒ (x - 1)2 ≤ 0 ⇒ -(x- 1)2 + 10 ≤ 10. ⇒ f(x) ≤ 10. ∴maximum value of f(x) = 10 when x - 1 = 0 ⇒ x = 1. And minimum value of f(x) does not exist. (iv) g(x) = x3 + 1 A(iv) we have, g(x) = x3 + 1 For the given fxn, x∈?, g(x)→∞ as x→∞ and g(x)−∞ as x→−∞ Q Maximum and minimum value does not exist.

Updated on Aug 4, 2025 17:05 IST

By Pallavi Pathak, Assistant Manager Content

Application of Derivatives Class 12 covers how the derivatives can be used to find the equations of tangent and normal to a curve at a point, to determine the rate of change of quantities, and to find turning points on the graph of a function. The chapter covers the applications of the derivative in various disciplines such as Science, Engineering, Social Sciences, and other fields.
Application of Derivatives Class 12 NCERT Solutions covers the increasing and decreasing functions, the rate of change of quantities, and maxima and minima. The subject matter experts at Shiksha have created the NCERT solution to help students deepen their understanding of concepts. It is ideal for exam preparation. The detailed and step-by-step solutions help students to score high in the CBSE Board exam and competitive exams like JEE Mains.
To get the NCERT Solutions of Class 11 and 12 of all three subjects, including Chemistry, Maths, and Physics, check - NCERT Solutions Class 11 and 12.

Table of content

Quick Overview of Applications of Derivatives – Class 12 Maths
Application of Derivatives NCERT Solution PDF: Free Download
Class 12 Maths Application of Derivatives: Key Topics, Weightage
Important Formulas of Application of Derivatives Class 12
Class 12 Application of Derivatives Exercise 6.1 Solutions
NCERT Solutions for Class 12 Maths Ch 6 Application of Derivatives- FAQs
Class 12 Application of Derivatives 6.5 Solutions- (Old NCERT)
NCERT Class 12 Application of Derivatives Miscellaneous Exercise Solutions

Quick Overview of Applications of Derivatives – Class 12 Maths

Here is a summary of the Applications of Derivatives Class 12 NCERT Solutions:

The chapter covers the rate of change of quantities and the chain rule.
It also includes the First Derivative Test, comprising the concepts of local maxima, local minima, and point of inflexion.
It covers the Second Derivative Test and the working rule of finding absolute maxima and/or absolute minima.

To get access to the NCERT Solutions of all chapters of the Class 12 Maths with free PDF, and important topics, check - Class 12 Maths NCERT Solutions.

Class 12 Maths Application of Derivatives: Key Topics, Weightage

Application of Derivatives Class 12 is a part of calculus in JEE Main exams, which is a significant area of the Mathematics section. See below the topics covered in this chapter:

Exercise	Topics Covered
6.1	Introduction
6.2	Rate of Change of Quantities
6.3	Increasing and Decreasing Functions
6.4	Maxima and Minima

Application of Derivatives Class 12 Weightage in JEE Mains

Exam	Number of Questions
JEE Main	2-4 questions

Important Formulas of Application of Derivatives Class 12

Application of Derivatives: Important Formulae for CBSE and Competitive Exams

Rate of Change: $\frac{dy}{dx}$ , where $y$ is the dependent variable and $x$ is the independent variable.
Slope of Tangent: $m = f'(x)$
Slope of Normal: $m = -\frac{1}{f'(x)}$
Equation of the tangent: $y - y_1 = f'(x_1)(x - x_1)$
Equation of the normal: $y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)$
Approximation Formula: $f(x + h) \approx f(x) + h \cdot f'(x)$
Condition for Increasing Function: $f'(x) > 0$
Condition for Decreasing Function: $f'(x) < 0$
Local Maxima/Minima (First Derivative Test): Solve $f'(x) = 0$ and analyze sign changes of $f'(x)$ around the critical points.
Conditions for Minima and Maxima:
- - $f'(x) = 0$ and $f''(x) > 0$ for local minima.
  - $f'(x) = 0$ and $f''(x) < 0$ for local maxima.
Second Derivative Test: Use $f''(x)$ to confirm the nature of critical points.

Class 12 Application of Derivatives Exercise 6.1 Solutions

Class 12 Application of Derivatives Exercise 6.1 discusses the fundamental usage of Derivative techniques to solve various kinds of problems. Application of Derivatives Exercise 6.1 focuses on the rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and finding maxima and minima. The concepts discussed in Class 12 Math Exercise 6.1 Solutions will help students get a good understanding of calculus and even provide tools to solve problems in Class 12 physics and other fields too. Class 12 Ch 6 Exercise 6.1 Solutions includes 18 Questions (10 Long, 6 Short, 2 MCQs). Students can check the complete solutions for Exercise 6.1 below;

Application of Derivatives Exercise 6.1 Solutions

Q1. Find the rate of change of the area of a circle with respect to its radius r when

$\frac{d}{d r} = \frac{d}{d r}$ (πr²) = 2πr.

A.1. (a) r = 3cm

When r = 3 cm,

$\frac{d}{d r}$ = 2 × π × 3 cm = 6π.

Thus, the area of the circle is changing at the rate of 6π cm $c m^{\frac{2}{5}}$ .

(b) r = 4cm

whenr = 4cm,

$\frac{d}{d r}$ = 2 × π × 4cm = 8π.

Thus, the area of the circle is changing at the rate of 8 $c m^{\frac{2}{5}}$ .

Q2. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of an edge is 12 cm?

A.2. Let x be the length of edge,v be the value and s be the surface area of the cube then,

y = x3.

and S = 6x2, where x is a fxn of time.

Now, $\frac{d Y}{d F} = 8 c m^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d t}$ (x3) = 8

$\frac{d x^{3}}{d x} \cdot \frac{d x}{d t} = 8$ (by chain rule)

$\Rightarrow$ 3x2 $\frac{d x}{d x} = 8 .$

$\frac{d x}{d t} = \frac{8}{3 x^{2} .}$

Now, $\frac{d S}{d t} = \frac{d}{d t}$ (bx2) = $\frac{d (6 x^{2})}{d x} - \frac{d x}{d x}$ = 12x $\frac{8}{3 x^{2}} = \frac{32}{x}$ $c m^{\frac{2}{5}}$ .

When x = 12 cm,

$\frac{d S}{d t} = \frac{32}{12} = \frac{8}{3}$ $c m^{\frac{2}{5}}$ .

Q3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

A.3. Let ‘r’ cm be the radius of the circle which is afxn of time.

Then, $\frac{d y}{d t}$ = 8 3cm/s as it is increasing.

Now, the area A of the circle is A = πr2.

So, the rate at which the area of the circle change $\frac{d}{d t} = \frac{d}{d t}$ πr2.

$= \frac{d}{d r} π r^{2} \cdot \frac{d r}{d t}$

= 2πr 3

= 6πr. ${cm}^{\frac{2}{5}}$

When r = 10cm,

$\frac{d}{d t}$ = 6.π × 10 = 60π ${cm}^{\frac{2}{5}}$

Q4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

A.4. Let ‘x’ cm be the length of edge of the cube which is a fxⁿ of time t then,

$\frac{d x}{d t}$ = 3cm/s as it is increasing.

Now, volume v of the cube is v = x³

Ø Rate of change of volume of the cube $\frac{d v}{d t} = \frac{d q^{3}}{d t} .$

$= \frac{d x^{3}}{d x} \frac{d x}{d t}$

= 3x².3

= 9x² ${cm}^{\frac{2}{5}}$

When x = 10cm.

$\frac{d v}{d t}$ = 9 x (10)²= 900 ${cm}^{\frac{2}{5}}$

Commonly asked questions

27. Prove that $y = \frac{4 s i n θ}{(2 + c o s θ)} - θ$ is an increasing function of θ $(0, \frac{π}{2})$ .

We have, y = $\frac{4 s i n θ}{(2 + c o s θ)} - θ$

Differentiating w rt.Ø we get,

$\frac{d y}{d θ} = \frac{d}{d θ} [\frac{4 s i n θ}{2 + c o s θ} - θ]$

$= \frac{(2 + c o r θ) \frac{d}{d θ} (4 s i n θ) - 4 s i n θ \frac{d}{d θ} (2 + c o r θ)}{{(2 + c o s θ)}^{2}} - \frac{d θ}{d θ} .$

$= \frac{(2 + c o s θ) (4 c o s θ) - 4 s i n θ (- s i n θ)}{{(2 + c o s θ)}^{2}} - 1$

$= \frac{8 c o s θ + 4 c o s^{2} θ + 4 s i n^{2} θ \cdot}{{(2 + c o s θ)}^{2} .} - 1$

$= \frac{8 c o s θ + 4 (c o s^{2} θ + s i n^{2} θ) - {(2 + c o s θ)}^{2}}{{(2 + c o s θ)}^{2}}$

$= \frac{8 c o r θ + 4 - [4 + c o s^{2} θ + 4 c o s θ]}{{(2 + c o r θ)}^{2}} .$

$= \frac{8 c o s θ + 4 - 4 - c o s^{2} θ - 4 c o s θ}{{(2 + c o s θ)}^{2}}$

$= \frac{4 c o s θ - c o s^{2} θ}{{(2 + c o s θ)}^{2}} = \frac{c o s θ (4 - c o s θ)}{{(2 + c o s θ)}^{2}}$

When $θ \in [0, \frac{π}{2}]$ we know that, $0 \leq c o s θ \leq 1 .$

So, $4 - c o s θ > 0$

And also, (2 + cos $θ$ )²> 0.

$∴ \frac{d y}{d θ} \geq 0 \forall θ \in [0, \frac{π}{2}]$

Hence, y is an increasing fxⁿ of $θ$ in $[0, \frac{π}{2}]$

74.Find the maximum and minimum values, if any, of the following functions

given by.

(i) $f (x) = {(2 x - 1)}^{2} + 3$

(i) We have, f(x) = (2x - 1)² + 3.

For all $x \in ?, (2 x - 1)^{2} \geq 0$

(2x - 1)² + 3 ≥ 3.

f(x) ≥ 3.

∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = $\frac{1}{2}$

Again as $x \to \infty, f (x) \to \infty$ as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values.

(ii) $f (x) = 9 x^{2} + 12 x + 2$

A(ii)

We have, f(x) = 9² + 12x + 2.

$f (x) = 9 [x^{2} + \frac{12 x}{9} + \frac{2}{9}]$ (Taking 9 common from each team).

$f (x) = 9 [x^{2} + \frac{4 x}{3} + \frac{2}{9}]$

$f (x) = 9 [x^{2} + 2 \times \frac{2 x}{3} + {(\frac{2}{3})}^{2} - {(\frac{2}{3})}^{2} + \frac{2}{9}]$

$f (x) = 9 [{(x + \frac{2}{3})}^{2} - \frac{4}{9} + \frac{2}{9}]$

$f (x) = 9 [\cdot {(x + \frac{2}{3})}^{2} - \frac{2}{9}] = 9 {(x + \frac{2}{3})}^{2} - 2$

For all $x \in ?, {(x + \frac{2}{3})}^{2} \geq 0$

$\Rightarrow {(x + \frac{2}{3})}^{2} - 2 \geq \cdot 0 - 2$

$\Rightarrow$ f(x)≥ - 2.

∴The minimum value of f(x) = -2 when $x + \frac{2}{3} = 0$

And as $x \to \infty, f (x) \to \infty$ so f(x) has $\Rightarrow x = - \frac{2}{3} .$

no maximum values.

(iii) f(x) = (x - 1)² + 10

A(iii)

we have, f(x) = - (x - 1)² + 10

For all $x \in ?, (x - 1)^{2} \geq 0 .$

$\Rightarrow$ (x - 1)² ≤ 0

$\Rightarrow$ -(x- 1)² + 10 ≤ 10.

$\Rightarrow$ f(x) ≤ 10.

∴maximum value of f(x) = 10 when x - 1 = 0 $\Rightarrow$ x = 1.

And minimum value of f(x) does not exist.

(iv) g(x) = x³ + 1

A(iv)

we have, g(x) = x³ + 1

For the given fxⁿ, $x \in ?,$

$g (x) \to \infty a s x \to \infty$

and $g (x) - \infty a s x \to - \infty$

Q Maximum and minimum value does not exist.

14. Sand is pouring from a pipe at the rate of 12 cm³ /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Let r cm and h cm be the radius and the height of the cone. Then,

h = $\frac{1}{6}$ r. H = 6h

So, volume, V of the cone = $\frac{1}{3}$ πr²h

$= \frac{1}{3} π {(6 h)}^{2} \times h .$ $\frac{4 d}{d t} .$

= 12 × h³

Rate of change of volume of the cone wrt the height is

$\frac{dV}{d h} = \frac{d}{d h} (12 π h^{3})$ = 12 × π × 3 × h².

As the sand is pouring from the pipe at rate of 12 ${cm}^{\frac{3}{5}}$

we have

$\frac{d}{d t} = 12$

$\Rightarrow \frac{d v}{d h} \times \frac{d h}{d t} = 12$

$\Rightarrow 36 π h^{2} \frac{d h}{d t} = 12 .$

$\Rightarrow \frac{d h}{d t} = \frac{12}{36 π h^{2}} = \frac{1}{3 π h^{2}} .$

${∴ \frac{d h}{d t} |}_{h = 4} = \frac{1}{3 π \times {(4)}^{2}} = \frac{1}{48 π} .$

Hence, the height is increasing at the rate of $\frac{1}{48 x}$ cm/s.

63. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is

(A) 3 (B) $\frac{1}{3}$ (C) -3 (D) - $\frac{1}{3}$

Given, $y = 2 x^{2} + 3 s i n x$

Slope of tangent, $\frac{d y}{d x} = 4 x + 3 c o s x$

${\frac{d y}{d x} |}_{x = 0} = 4 \times 0 + 3 \times c o s 0 = 3$

So, slope of normal $= \frac{- 1}{3}$

Option (D) is correct.

99. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $s i n^{- 1} \frac{1}{3}$ .

Kindly go through the solution

122. The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is

(A) 1 (B) 2 (C) 3 (D) $\frac{1}{2}$

The equation of the tangent to the given curve is $y = m x + 1 .$

Now, substituting $y = m x + 1 .$ in $y^{2} = 4 x,$ we get:

$\begin{array}{l} \Rightarrow {(m x + 1)}^{2} = 4 x \\ \Rightarrow m^{2} x^{2} + 1 + 2 m x - 4 x = 0 \\ \Rightarrow m^{2} x^{2} + x (2 m - 4) + 1 = 0 . . . . . . . . . . (i) \end{array}$

Since a tangent touches the curve at one point, the roots of equation (i) must be equal.

Therefore, we have:

Discriminant = 0

$\begin{array}{l} {(2 m - 4)}^{2} - 4 {(m)}^{2} (1) = 0 \\ \Rightarrow 4 m^{2} + 16 - 16 m - 4 m^{2} = 0 \\ \Rightarrow 16 - 16 m = 0 \\ \Rightarrow m = 1 \end{array}$

Hence, the required value of m is 1.

Therefore, option (A) is correct.

21. Show that the function given by f (x) = sin x is (a)

increasing in $(0, \frac{π}{2})$ (b) decreasing in $(\frac{π}{2}, π)$

(c) neither increasing nor decreasing in $(0, π)$

We have, f (x) = sin x.

So, f (x) = cosx.

(a) when, x ∈ $(0, \frac{π}{2})$ i e, x in 1st quadrat.

f (x) = cos.x> 0

f (x) is strictly increasing in $(0, \frac{π}{2})$ .

(b) when, x ∈ $(\frac{π}{2}, π)$ in IInd quadrat

f (x) = cosx< 0.

∴f (x) is strictly decreasing (π

f (x) = cosx is increasing in $(0, \frac{π}{2})$ and decreasing

in $(\frac{π}{2}, π)$ and f $(\frac{π}{2})$ = cos $\frac{π}{2} = 0 .$

∴f (x) is neither increasing not decreasing in (0, π).

32. For what values of a the function f given by f(x) = x² + ax + 1 is increasing on [1, 2]?

We have, f (x) = x² + x + 1

So, f (x) = 2x + a

If f (x) is strictly increasing onx $(1, 2), f^{'} (x) > 0 .$

So, 1

$\to 2 \times 1 < 2 \times x < 2 \times 2$

$\Rightarrow 2 < 2 x < 4$

$\Rightarrow 2 + a < 2 x + a < 4 + a .$

$\Rightarrow 2 + a < f^{'} (x) < 4 + a .$

The minimum value of f (x) is 2 + a and that of men value is 4 + a.

∴ 2 +a> 0and 4 + a> 0

$\Rightarrow$ a> -2 and a> -4.

$\Rightarrow$ a> -2.

∴The best value of a is -2.

15. The total cost C (x) in Rupees associated with the production of x units of an item is given by C (x) = 0.007x³ – 0.003x² + 15x + 4000.

Find the marginal cost when 17 units are produced.

Given, c (x) = 0.007 x³- 0.003x² + 15x + 400.

Since the marginal cost is the rate of change of total cost wrt the output we have,

Marginal cost, MC, = $\frac{dC}{d x} (x)$

= 0.007 * 3x²- 0.003 * 2x + 15.

When x = 17,

Then, MC = 0.007 * 2. (17)² - 0.003 2 (17) + 15.

= 6.069 - 0.102 + 15.

= 20.967

Hence, the required marginal cost = ' 20, 97.

33. Let I be any interval disjoint from [–1, 1]. Prove that the function f given by

$f (x) = x + \frac{1}{x}$ is increasing on I.

We have, f (x) = $\overset{?}{x} + \frac{1}{x}$

So, f (x) = $1 - \frac{1}{x^{2}} = \frac{x^{2} - 1}{x^{2}} = \frac{(x - 1) (x + 1)}{x^{2}}$

So, for every x∈ I, where I is disjoint from [-1,1]

f (x) = $\frac{(+ v e) (+ v e)}{(+ v e)} = (+ v e) > 0 w h e n x > 1 .$

and f (x) = $\frac{(- v e) (- v e)}{(+ v e)}$ = ( + ve) > 0 when x< -1.

∴f (x) is strictly increasing on I .

44. Find points at which the tangent to the curve y = x³ – 3x² – 9x + 7 is parallel to the x-axis.

The given eqⁿ of the curve is $y = x^{3} - 3 x^{2} - 9 x + 7 .$

slope of tangent to the given curve, $\frac{d y}{d x} = 3 x^{2} - 6 x - 9$

when the tangent is parallel to x-axis $\frac{d y}{d x} = 0$

$\Rightarrow 3 x^{2} - 6 x - 9 = 0$

$\Rightarrow x^{2} - 2 x - 3 = 0$

$\Rightarrow x^{2} + x - 3 x - 3 = 0$

$\to x (x + 1) - 3 (x + 1) = 0$

$\to (x + 1) (x - 3) = 0$

x = 3 or x = -1

When x = 3, $y = 3^{3} - 3 {(3)}^{2} - 9 (3) + 7 = 27 - 27 - 27 + 7 = - 20$

And when x = -1 $y = {(- 1)}^{3} - 3 {(- 1)}^{2} - 9 (- 1) + 7 = - 1 - 3 + 9 + 7 = 12$

Hence, the required points are $(3, - 20) (- 1, 12)$

22. Find the intervals in which the function f given by f(x) = 2x² – 3x is

(a) Increasing (b) Decreasing

We have, f (x) = 2x² 3x

So, f (x) = $\frac{d}{d x} (2 x^{2} - 3 x) = 4 x - 3 .$

Atf (x) = 0.

$\Rightarrow$ 4x - 3 = 0

i e, $x = \frac{3}{4}$ divides the real line into two

disjoint interval $(- \infty, \frac{3}{4}) (\frac{3}{4}, \infty)$

(a) Now,

f (x) = 4x - 3 > 0 $\forall x \in (\frac{3}{4}, \infty)$

So, f (x) is strictly increasing in $(\frac{3}{4}, \infty)$

(b) Now, f (x) = 4x - 3 < 0 $\forall x (- \infty, \frac{3}{4})$

So, f (x) is strictly decreasing in $(- \infty, \frac{3}{4})$

23. Find the intervals in which the function f given by f(x) = 2x³ – 3x² – 36x + 7 is

(a) Increasing (b) Decreasing

We have, f (x) = 2x³- 3x²- 36x + 7.

So, f (x) = $\frac{d}{d x} (2 x^{3} - 3 x^{2} - 36 x + 7) = 6 x^{2} - 6 x - 36 .$

= 6 (x²-x- 6).

= 6 (x²- 3x + 2x- 6)

= 6 [x (x- 3) + 2 (x- 3)]

= 6 (x- 3) (x + 2).

At, f (x) = 0

$\Rightarrow$ 6 (x- 3) (x + 2) = 0.

So, when x- 3 = 0 or x + 2 = 0.

$\Rightarrow$ x = 3 or x = -2.

Hence we an divide the real line into three disjoint internal

I $(- \infty, - 2) II (- 2, 3)andIII (3, \infty)$

At x ∈ $(- \infty, - 2),$

f (x) = ( + ve) ( -ve) ( -ve) = ( + ve) > 0.

So, f (x) is strictly increasing in $(- \infty, - 2)$

At, x∈ ( -2,3),

f (x) = ( + ve) ( + ve) ( -ve) = ( -ve) < 0.

So, f (x) is strictly decreasing in ( -2,3).

At, x ∈ $(3, \infty)$

f (x) = ( + ve) ( + ve) ( + ve) = ( + )ve> 0.

So, f (x) is strictly increasing in $(3, \infty)$

∴ (a) f (x) is strictly increasing in $(- \infty, - 2)and (3, - \infty)$

(b) f (x) is strictly decreasing in ( -2,3)

19. Show that the function given by f (x) = 3x + 17 is increasing on R.

We have, .

f (x) = 3x + 17.

So, f (x) = 3 > 0 $\forall x \in R$

∴f (x) is strictly increasing on R.

20. Show that the function given by f (x) = e2x is increasing on R.

We have, f (x) = e^2x

So, f (x) = $\frac{d}{d x} e^{2 x}$ = $e^{2 x} \frac{d}{d x} 2 x$ = 2e^2x> 0 $\forall x \in R .$

∴f (x) is strictly increasing on R.

25. Show that 2 log(1 ) 2 x y x x = +− + , x > – 1, is an increasing function of x throughout its domain.

We have, y = log $(1 + x) - \frac{2 x}{2 + x}, x > - 1$

Differentiating the above wrt.x.we get,

$\frac{d y}{d x} = \frac{1}{1 + x} \frac{d}{d x} (1 + x) -$ $\frac{(2 + x) \frac{d}{d x} 2 x - 2 x \frac{d}{d x} (2 + x)}{{(2 + x)}^{2}}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{1 + x} - \frac{(2 + x) \cdot 2 - 2 x}{{(2 + x)}^{2}} .$

$\Rightarrow \frac{d y}{d x} = \frac{1}{1 + x} - \frac{4 + 2 x - 2 x}{{(2 + x)}^{2}} = \frac{1}{1 + x} - \frac{4}{{(2 + x)}^{2}} .$

$= \frac{{(2 + x)}^{2} - 4 (1 + x)}{(1 + x) {(2 + x)}^{2}}$

$= \frac{4 + x^{2} + 4 x - 4 - 4 x}{(1 + x) {(2 + x)}^{2}}$

$\frac{d y}{d x} = \frac{x^{2}}{(1 + x) {(2 + x)}^{2}} .$

The given domain of the given function isx> -1.

$\Rightarrow$ (x + 1) > 0.

Also, $x^{2} \geq 0$

(2 + x)²> 0.

Hence, $\frac{d y}{d x} = \frac{(+ v e)}{(+ v) (+ v e)} =(+ve) > 0 .$

∴ y is an increasing function of x throughout its domain.

24. Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x² + 2x – 5 (b) 10 – 6x – 2x² (c) –2x³ – 9x² – 12x + 1 (d) 6 – 9x – x²

(e) (x + 1)³ (x – 3)³

(a) f (x) = x² + 2x - 5.

f(x) = 2x + 2 = 2 (x + 1).

At, f(x) = 0

2 (x + 1) = 0

$\Rightarrow$ x = -1.

At, x $(- \infty, - 1),$

f(x) = (- ) ve< 0.

So, f (x)is strictly decreasing or $(- \infty, - 1) .$

At x ∈ $(- 1, \infty)$

f(x) = ( + ve) > 1

f(x) is strictly increasing on $(- 1, \infty) .$

(b) f(x) = 10 - 6x- 2x²

So, f(x) = - 6 - 4x = - 2 (3 + 2x).

Atf(x) = 0

$\Rightarrow$ 2 (3 + 2x) = 0.

$\Rightarrow$ x = $\frac{- 3}{2}$

At x $(- \infty, \frac{- 3}{2}),$

∴f(x) is strictly increasing on $(- \infty, \frac{- 3}{2})$

At x $(\frac{- 3}{2}, \infty)$

f(x) = ( -ve) ( + ve) = ( - ) ve< 0.

∴f (x) is strictly decreasing on $(\frac{- 3}{2}, \infty)$

(c) f (x) = 2x³- 9x²- 12x.

So, f (x) =- 6x²- 18x- 12 = - 6 (x² + 3x + 2).

= -6 [x² + x + 2x + 2]

= -6 [x (x + 1) + 2 (x + 1)]

= -6 (x + 1) (x + 2)

At, f (x) = 0.

$\Rightarrow$ 6 (x + 1) (x + 2) = 0

$\Rightarrow$ x = -1 and x = -2.

At, x $(- \infty, - 2)$ , f(x) = ( -ve) ( -ve) ( -ve) = ( -ve) < 0

So, f(x) is strictly decreasing.

At, x( -2, -1), f(x) = ( -ve) ( -ve) ( -ve) = ( -ve) < 0.

So, f(x) is strictly decreasing

(d) f(x) = 6 - 9x-x²

So, f(x) = - 9 - 2x

At, f(x) = 0

$\Rightarrow$ 9 - 2x = 0

$\Rightarrow$ x = $\frac{- 9}{2}$

At, x ∈ $(- \infty, \frac{- 9}{2}),$

f(x) > 0

So, f(x) is strictly increasing.

At x ∈ $(\frac{- 9}{2}, \infty) .$

f(x) < 0.

So, f(x) is strictly decreasing.

(e) f(x) = (x + 1)³ (x- 3)³.

So, f(x) = (x + 1)³ $\frac{d}{d x} {(x - 3)}^{3} + {(x - 3)}^{3} \frac{d}{d x} {(x + 1)}^{3}$

= (x + 1)³. 3 (x- 3)² + (x- 3)³. 3 (x + 1)²

= 3 (x + 1)² (x- 3)² (x + 1 + x- 3).

= 3 (x + 1)² (x- 3)² (2x- 2).

= 6 (x + 1)² (x- 3)² (x- 1).

For strictly increasing,

f(x) > 0.

As 6 > 0,(x + 1)²> 0 and (x- 3)²> 0. We need

(x 1) > 0.

$\Rightarrow$ x> 1.

∴f (x) is strictly increasing on $(1, \infty) .$

And strictly decreasing on $(- \infty, 1) .$

42. Find the slope of the normal to the curve $x = a c o s^{3} θ, y = a s i n^{3} θ, a t, θ = \frac{π}{4}$

The Equation of the given curve are

$x = a c o s^{3} θ y = a s i n^{3} θ$

So, $\frac{d x}{d θ} = 3 a c o s^{2} θ (- s i n θ) = - 3 a c o s^{2} θ s i n θ .$

and $\frac{d y}{d θ} = 3 a s i n^{2} θ c o s θ$

$∴ \frac{d y}{d x} = \frac{d y / d θ}{d x / d θ} = \frac{3 a s i n^{2} θ c o s θ}{- 3 a c o s^{2} θ s i n θ} = - t a n θ$

So, ${\frac{d y}{d x} |}_{x = π / 4} = - t a n \frac{π}{4} = - 1$ which is the slope of the tanget to the curve.

Now, required slope of normal to the curve = $- 1$
$Slopeoftangent$
$= \frac{- 1}{- 1} = 1$

92. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Let A.B.C.D.be the square increased in a given fixed circle with radius x

Let ‘x’ and ‘y’ be the length and breadth of the rectangle

∴x, y> 0

In ABC, right angle at B,

x² + y² = (2x)²

$\Rightarrow$ x² + y² = 4x²

30. Which of the following functions are decreasing on $(0, \frac{π}{2})$ ?

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

(A) We have,

f(x) = cosx

So, f(x) = -sin x

When x $\in [0, \frac{π}{2})$ we know that sin x> 0.

$\Rightarrow$ -sinx< 0. f(x) < 0 $\forall x \in (0, \frac{π}{2})$

∴f(x) is strictly decreasing on $(0, \frac{π}{2})$

(B) We have, f(x) = cos 2x

So, f(x) = -2sin 2x

When $x \in (0, \frac{π}{2})$ we know that sin x> 0.

i e, 0 $\frac{π}{2}$
=> 0<2x<π

So, sin 2x> 0 (sinØ is ( +ve) in 1st and 2ndquadrant).

$\Rightarrow$ -2sin 2x< 0.

$\Rightarrow$ f (x) < 0.

∴f (x) is strictly decreasing on $(0, \frac{π}{2}) .$

(c) We have, f(x) = cos 3x

$\Rightarrow$ f(x) = -3 sin 3x.

As $0 < x < \frac{π}{2}$

$\to 0 < 3 x < \frac{3 π}{2}$

We can divide the interval into two

Case I At, 0 < 3x

sin 3x> 0.

$\Rightarrow$ -3 sin 3x< 0 ${\begin{array}{l} ? 0 < B x < π \end{array}$

$\begin{array}{l} \to 0 < x < \frac{π}{3} . \end{array}$

$\Rightarrow$ f(x) < 0.

∴f(x) is strictly decreasing on $(0' \frac{π}{3})$

case II. At $π < 3 x < \frac{3 π}{2} w e g e t (i i i^{r d} q u a d r o u t) .$

sin 3x< 0.

$\Rightarrow$ -3 sin 3x> 0.

$\Rightarrow$ f(x) > 0. ${? π < 3 x < \frac{3 π}{2} \Rightarrow \frac{π}{3} < x < \frac{π}{2}}$

∴f(x) is strictly increasing on $(\frac{π}{3}, \frac{π}{2})$

Hence, f(x) is with a increasing or decreasing on $(0, \frac{π}{2}) .$

(d) We have, f(x) = tan x.

So, f(x) = sec²x.

$\Rightarrow$ f(x) = sec²x> 0 $\forall x \in (0, \frac{π}{2}) .$

∴f(x) is strictly increasing on $(0, \frac{π}{2}) .$

Hence, the fxⁿcosx and cos 2x are strictly decreasing on $(0, \frac{π}{2}) .$

97. Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.

Let r and h be the radius and height of the cone.

The volume V of the cone is.

$= \frac{1}{3} π r^{2} h$

$r^{2} h = \frac{3V}{π} = k (SAY) \Rightarrow r^{2} = \frac{k}{h} .$

And curve surface area S is

1. Find the rate of change of the area of a circle with respect to its radius r when

$\frac{d}{d r} = \frac{d}{d r}$ (πr²) = 2πr.

(a) r = 3cm

When r = 3 cm,

$\frac{d}{d r}$ = 2 × π × 3 cm = 6π.

Thus, the area of the circle is changing at the rate of 6π cm $c m^{\frac{2}{5}}$ .

(b) r = 4cm

whenr = 4cm,

$\frac{d}{d r}$ = 2 × π × 4cm = 8π.

Thus, the area of the circle is changing at the rate of 8 $c m^{\frac{2}{5}}$ .

31. On which of the following intervals is the function f given by f(x) = x¹⁰⁰ + sin x –1 decreasing ?

(A) (0,1) (B) $(\frac{π}{2}, π)$ (C) $(0, \frac{π}{2})$ (D) None of these

We have, f (x) = x 100 + sin x - 1.

So, f (x) = 100x⁹⁹ + cosx.

(A) When x (0,1). We get.

x>0

$\Rightarrow$ x⁹⁹> 0

$\Rightarrow$ 100 x⁹⁹> 0.

Now, 0 radian = 0 degree

and 1 radian = $\frac{18 0^{°} \times 1}{π} = \frac{18 0^{°}}{(\frac{22}{7})} = 5 7^{°} .$

So, cosx> 0 for x∈ (0,1) radian = (0,57)

∴f (x) > 0 for x∈ (0,1).

(B) When x ∈ $(\frac{π}{2}, π)$ we get,

So, x> 1

$\Rightarrow$ x⁹⁹> 1

$\Rightarrow$ 100x⁹⁹> 100. ${\begin{matrix} ? (\frac{π}{2}, π) = (1.57, 3.14) & which is great than 1 \end{matrix}}$

And cosx is negative between -1 and 0.

So, f (x) = 100x⁹⁹ + cosx> 100 - 1 = 99 > 0.

∴f(x) is strictly increasing on $(\frac{π}{2}, π)$

(c) When x ∈ $(0, \frac{π}{2}),$ we get,

$\Rightarrow$ x> 0

$\Rightarrow$ x⁹⁹> 0

$\Rightarrow$ 100x⁹⁹> 0

and cosx> 0. (firstquadrant).

I e, f(x) > 0.

∴f(x) is strictly increasing on $(0, \frac{π}{2})$

Hence, option (D) is correct.

Q38. Find the slope of the tangent to the curve y = 3x⁴ – 4x at x = 4.

The given eqⁿ of the curve is $y = 3 x^{4} - 4 x .$

Slope of the tangent at x = 4 is given by

${{\frac{d y}{d x}]}_{x = 4} = 12 x^{3} - 4]}_{x = 4} = 12 {(4)}^{3} - 4 = 12 \times 64 - 4$

$= 768 - 4$

= 764

62. Find the equation of the tangent to the curve y = √3x - 2 which is parallel to the line 4x - 2y + 5 = 0.

18. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

(A) 116 (B) 96 (C) 90 (D) 126

Given, R (x) = 3x² + 36x + 5.

Marginal revenue, $\frac{d}{d x} R (x) = \frac{d}{d x} (3 x^{2} + 36 x + 5)$

= 3 * 2x + 36

= 6x + 36

When x = 15.

$\frac{d}{d x} R (x) = 6 * 15 + 36$ = 90 + 36 = 126

Option (D) is correct.

29. Prove that the function f given by f(x) = x² – x + 1 is neither strictly increasing nor decreasing on (– 1, 1).

We have, f (x) = x²-x + 1.

So, f (x) = $\frac{d}{d x} (x^{2} - x + 1) = 2 x - 1$

Atf (x) = 0.

$\Rightarrow$ 2x - 1 = 0

I e, x = $\frac{1}{2}$ divides the real line into two disjoint interval the interval ( -1, 1) into

Two disjoint interval

$(- 1, \frac{1}{2}) (\frac{1}{2}, 1)$

And f (x) is strictly increasing in $(\frac{1}{2}, 1)$ and strictly decreasing in $(- 1, \frac{1}{2})$

Hence, f (x) is with a increasing or decreasing on ( -1,1).

11. A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate

Given eqⁿ of the curve is 6y = x³ + 2.______ (1)

Wheny coordinate change s 8 times as fast as x-coordinate

$\frac{d y}{d x}$ = 8 _____ (2)

Now, differentiating eqⁿ (1) wrt.x we get,

$\frac{d}{d x} (6 y) = \frac{d}{d x} (x^{3} + 2)$

$\Rightarrow 6 \frac{d y}{d x} = 3 x^{2} + 0 .$

6 × 8 = 3x² (using eqⁿ (2)

$\Rightarrow x^{2} = \frac{48}{3} = 16$

$\Rightarrow +\sqrtx = \pm\sqrt16$

x = $\pm$ 4.

When x = 4, we have, 6y = 4³+ 2 = 64 + 2 + 66

$\Rightarrow y = \frac{66}{6}$ y =11.

And when x = -4, we have, 6y = ( -4)3 + 2 = -64 + 2 = -62

$\Rightarrow y = \frac{62}{6} = \frac{31}{3}$

The tequired point s are (4, 11) and $(- 4, \frac{31}{3})$

13. A balloon, which always remains spherical, has a variable diameter 3 (2 1) 2 x + . Find the rate of change of its volume with respect to x.

Given, diameter of the spherical balloon = $\frac{3}{2}$ (2x + 1)

So, radius of the spherical r = $\frac{1}{2} \times \frac{3}{2} (2 x + 1)$

$= \frac{3}{4} (2 x + 1)$

Then, volume of the spherical V = $\frac{4}{9} π r^{3}$

$= \frac{4}{3} π \times [\frac{3}{4} {(\frac{3}{(2 x + 1)}]}^{3}$

$= \frac{9 π}{16} {(2 x + 1)}^{3} .$

Q Rate of change of volume wrt.tox, $\frac{d V}{d x} = \frac{d}{d x} [\frac{π}{16} {(2 x + 1)}^{3}]$

$= \frac{9 π}{16} \times 3 \times {(2 x + 1)}^{2} \cdot \frac{d}{d x} (2 x + 1)$

$= \frac{27 π}{16} {(2 x + 1)}^{2} \times 2 = \frac{27 π}{8} {(2 x + 1)}^{2} .$

26. Find the values of x for which y = [x(x – 2)]2 is an increasing function.

We have, y = [x (x- 2)]².

Differentiating the above w rt. x we get,

$\frac{d y}{d x} = \frac{d}{d x} [x] {(x - 2)}^{2}$

$= 2 [x (x - 2)] \frac{d}{d x} [x (x - 2)]$

$= 2 [x (x - 2)] [x \frac{d}{d x} (x - 2) + (x - 2) \frac{d x}{d x}]$

= 2 [x (x- 2)] (x + x- 2)

= 2x (x - 2) (2x - 2)

$\frac{d y}{d x}$ = 4x (x - 2) (x - 1).

Now, $\frac{d y}{d x} = 0$

$\Rightarrow$ 4x (x - 2) (x - 1) = 0.

i e, x = 0, x = 2, x = 1 divides the real line into

four disjoint interval. $(- \infty, 0],$ [0,1],[1,2] and $[2, \infty] .$

when x $[- \infty, 0] .$

$\frac{d y}{d x} = (- v e) (- v e) (- v e) =$ $(- v e) \leq 0, 0 x = 0$

∴f (x) is decreasing in $[- \infty, 0] .$

When $x \in [0, 1]$

$\frac{d y}{d x} = (+ v e) (- v e) (- v e)$ $= (+ v e) \geq 0, x = 0 x = 1$

∴f (x) is increasing in [0,1].

When x $\in [1, 2]$

$\frac{d y}{d x} = (+ v e) (- v e) (+ v e) = (- v e) \leq 0,$ $0for x = 1 + x = 2$

∴f (x) is decreasing.

When $x \in (2, \infty) .$

$\frac{d y}{d x} = (+ v e) (+ v e) (+ v e) = (+ v e) \geq 0, 0,for x = 2$

∴f (x) is in creasing

Hence, f (x) is increasing for x $\in [0, 1) x \in [2, \infty)$

28. Prove that the logarithmic function is increasing on (0, ∞).

We have, f (x) = log x.

So, f (x) = $\frac{d}{d x} (l o g x) = \frac{1}{x} .$

i e, f (x) $= \frac{1}{x}$ > 0. When $x \neq 0 x \in (0, \infty)$

Hence, the logarithmic fx is strictly increasing on $(0, \infty) .$

34. Prove that the function f given by f(x) = log sin x is increasing on $(0, \frac{π}{2})$ and decreasing on $(\frac{π}{2}, π)$ .

We have, f (x) = log sin x

So, f (x) = $\frac{1}{s i n x} \frac{d s i n x}{d x} = \frac{c o s x}{s i n x} = c o t x$

When $x \in (0, \frac{π}{2})$

f (x) = cot x> 0 (Ist quadrat )

So, f (x) is strictly increasing on $(0, \frac{π}{2})$

When x ∈ $(\frac{π}{2}, π)$

f (x) = cot x< 0. (IIndquadrant ).

So, f (x) is strictly decreasing on $(\frac{π}{2}, π)$

35. Prove that the function f given by f (x) = log |cos x| is decreasing on $(0, \frac{π}{2})$ and increasing on $(3 \frac{π}{2}, 2 π)$

We have, f (x) = log $| c o s x | .$

f (x) = $\frac{1}{c o s x} \frac{d}{d x} x = - \frac{s i n x}{c o s x} = - t a n x$

whenx $(0, \frac{π}{2}),$ we get.

tanx> 0 (Ist quadrant).

$\Rightarrow$ tanx< 0

$\Rightarrow$ f (x) < 0.

∴f (x) is decreasing on $(0, \frac{π}{2}) .$

When x ∈ $(\frac{3 π}{2}, 2 π)$ we get,

tanx|< |0 (ivth quadrant).

$\Rightarrow$ -tanx|>| 0

$\Rightarrow$ f (x) > 0

∴f (x) is increasing on $(\frac{3 π}{2}, 2 π) .$

37. The interval in which y = x² e^–x is increasing is (A) (– ∞, ∞) (B) (– 2, 0) (C) (2, ∞) (D) (0, 2)

We have, f (x) = x² e^–x

So, f (x) = $x^{2} \frac{d}{d x} e^{- x} + e^{- x} \frac{d x^{2}}{d x}$

$= x^{2} e^{- x} \frac{d}{d x} (- x) + e^{- x} 2 \cdot x \frac{d x}{d x}$

= -x² e^-x + e^-x 2x.

= x e^-x ( x + 2).

$= \frac{x (2 - x)}{e^{e x}}$

If f (x) = 0.

$\Rightarrow \frac{x (2 - x)}{e^{x}} = 0 .$

$\Rightarrow$ x = 0, x = 2.

Hence, we get there disjoint interval

$[- \infty, 0), (0, 2) (2, \infty)$

When, x $(- \infty, 0),$ we have, f (x) = ( -ve) ( + ve) = ( -ve) < 0.

So, f is strictly decreasing.

When x ∈ (0,2), f (x) = ( + ve) ( + ve) = ( + ve) > 0.

So, f is strictly increasing.

And when x ∈ $(2, \infty),$ f (x) = ( +ve) ( -ve) = ( -ve) < 0.

So, f is strictly decreasing.

Hence, option (D) is correct.

66. Find the approximate value of f(2.01), where f (x) = 4x² + 5x + 2.

Given, y = f (x) = 4x² + 5x + 2.

So, f (x) = 8x + 5. $\frac{d y}{d x}$ = 8x + 5 dy = (8x + 5) dx.

Let x = 2 and Δx = 0.01.Then,

f (x + Δx) = f (2 + 0.01) = f (2.01).

Δy = f. (x + Δx) f (Δx).

f (x +Δx) = f (x) +Δy.

= f (x) + dy = f (x) + (8x + 5) dx.

= f (2.01) = f (2) + (8 x 2 + 5). Δx {∴dx = Δx}

= 4 (2)² + 5 (2) + 2 + 21 (0.01)

= 16 + 10 + 2 + 0.21 = 28.21.

68. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.

We know that, the volume v of side ‘a’ mete of cube is v = x³.

So, $d = (\frac{d}{d x}) \cdot Δ x = 3 x^{2} \cdot Δ x .$

Given that, increase in side = 1% of x.

$\Rightarrow Δ x = \frac{x}{100}$

$∴ d v = 3 x^{2} (\frac{x}{100}) = 0.03 x^{3} m^{3} .$

82. What is the maximum value of the function sin x + cos x?

We have, f (x) = sin x + cos x.

f (x) = cos x - sin x.

At f (x) = 0

cosx - sin x = 0

sinx = cos x

$\Rightarrow \frac{s i n x}{c o s x} = 1 .$

$\Rightarrow t a n x = 1 = t a n \frac{π}{4}$

$\Rightarrow x = \frac{π}{4} o r x = n π + \frac{π}{4} .$

At $x = \frac{π}{4} + n π$ ,

$f (n π + \frac{π}{4}) = s i n (n π + \frac{π}{4}) + c o s (n x + \frac{π}{4}) .$

$= {(- 1)}^{x} s i n \frac{π}{4} + {(- 1)}^{n} s i n \frac{π}{4} .$

85. Find the maximum and minimum values of x + sin 2x on [0, 2π].

We have, f(x) = x + sin 2x ,x ∈ [0, 2π].

f(x) = 1 + 2cos 2x

At f(x) = 0

1 + 2 cos2x = 0

$\Rightarrow c o s 2 x = - \frac{1}{2} = - c o s \frac{π}{3} = c o s π - \frac{π}{3} = c o s \frac{2 π}{3}$

$∴ 2 x = 2 n π \pm \frac{2 π}{3}, n = 1, 2, 3 .$

$\Rightarrow x = n π \pm \frac{2 π}{3}$

$n = 0, x = \pm \frac{π}{3} \to x = \frac{π}{3} \in [0, 2 π] .$

$n = 1, x = π \pm \frac{π}{3} \to x = π + \frac{π}{3} a n d π - \frac{π}{3}$

$= \frac{4 π}{3} a d \frac{2 π}{3} \in [0, 2 π]$

$n = 2, x = 2 π \pm \frac{π}{3} \to x = 2 π + \frac{π}{3} a d 2 π - \frac{π}{3} .$

$\Rightarrow x = \frac{51}{3} \in [0, 2 π] .$

Hence, $x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3} a n d \frac{5 π}{3}$

Missing

At $x = \frac{π}{3}, f (\frac{π}{3}) = \frac{π}{3} + s i n \frac{2 π}{3} = 1.05 +$ $s i n (π - \frac{π}{3}) = 1.05 + s i n \frac{π}{3}$

$= 1.05 +\sqrt3/2$

= 1.05 + 0.87

= 1.92

At $x = \frac{2 π}{3}, f (\frac{2 π}{3}) = \frac{2 π}{3} + s i n 2 \times \frac{2 π}{3}$ $= 2.10 + s i n (π + \frac{π}{3})$

$= 2.10 - s i n \frac{π}{3} = 2, 10 - 0.87$

= 1.23

At $x = \frac{4 π}{3}, f (\frac{4 π}{3}) = \frac{4 π}{3} + s i n 2 \times \frac{4 π}{3} = 4 \cdot 2 + s i n (3 x - \frac{π}{3})$

$= 4.2 + s i n \frac{π}{3} = 4.2 + 0.87 .$

=5.07.

At $x = \frac{5 π}{3}, f (\frac{5 π}{3}) = \frac{5 π}{3} + s i n 2 \times \frac{5 π}{3} = 5.25 + s i n (3 x + \frac{π}{3})$

$= 5 \cdot 25 - s i n \frac{1}{3}$

= 5.25 - 0.87 = 4.38

At and points,

f(0) = 0 + sin2 × 0 = 0

f(2π) = 2π + sin 2 × 2π = 6.2 + 0 = 6.28

∴Maximum value of f(x) = 6.28 at x = 2π and

minimum value of f(x) = 0 at x= 0

120. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is $\frac{4}{27} π h^{3} t a n^{2} α$ .

Kindly go through the solution

123. The normal at the point (1,1) on the curve 2y + x² = 3 is (A) x + y = 0 (B) x – y = 0 (C) x + y +1 = 0 (D) x – y = 1

The equation of the given curve is $2 y + x^{2} = 3$ .

Differentiate with respect to x, we have:

$\begin{array}{l} \frac{2 d y}{d x} + 2 x = 0 \\ \Rightarrow \frac{d y}{d x} = - x \\ ∴ {\frac{d y}{d x}]}_{(1, 1)} = - 1 \end{array}$

The slope of the normal to the given curve at point (1,1) is

$\frac{- 1}{{\frac{d y}{d x}]}_{(1, 1)}} = - 1$

Hence, the equation of the normal to the given curve at (1,1) is given as:

$\begin{array}{l} \Rightarrow y - 1 = 1 (x - 1) \\ \Rightarrow y - 1 = x - 1 \\ \Rightarrow x - y = 0 \end{array}$

Therefore, option (B) is correct.

2. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of an edge is 12 cm?

Let x be the length of edge,v be the value and s be the surface area of the cube then,

y = x3.

and S = 6x2, where x is a fxn of time.

Now, $\frac{d Y}{d F} = 8 c m^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d t}$ (x3) = 8

$\frac{d x^{3}}{d x} \cdot \frac{d x}{d t} = 8$ (by chain rule)

$\Rightarrow$ 3x2 $\frac{d x}{d x} = 8 .$

$\frac{d x}{d t} = \frac{8}{3 x^{2} .}$

Now, $\frac{d S}{d t} = \frac{d}{d t}$ (bx2) = $\frac{d (6 x^{2})}{d x} - \frac{d x}{d x}$ = 12x $\frac{8}{3 x^{2}} = \frac{32}{x}$ $c m^{\frac{2}{5}}$ .

When x = 12 cm,

$\frac{d S}{d t} = \frac{32}{12} = \frac{8}{3}$ $c m^{\frac{2}{5}}$ .

3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Let ‘r’ cm be the radius of the circle which is afxn of time.

Then, $\frac{d y}{d t}$ = 8 3cm/s as it is increasing.

Now, the area A of the circle is A = πr2.

So, the rate at which the area of the circle change $\frac{d}{d t} = \frac{d}{d t}$ πr2.

$= \frac{d}{d r} π r^{2} \cdot \frac{d r}{d t}$

= 2πr 3

= 6πr. ${cm}^{\frac{2}{5}}$

When r = 10cm,

$\frac{d}{d t}$ = 6.π × 10 = 60π ${cm}^{\frac{2}{5}}$

4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Let 'x' cm be the length of edge of the cube which is a fxⁿ of time t then,

$\frac{d x}{d t}$ = 3cm/s as it is increasing.

Now, volume v of the cube is v = x³

Ø Rate of change of volume of the cube $\frac{d v}{d t} = \frac{d q^{3}}{d t} .$

$= \frac{d x^{3}}{d x} \frac{d x}{d t}$

= 3x².3

= 9x² ${cm}^{\frac{2}{5}}$

When x = 10cm.

$\frac{d v}{d t}$ = 9 x (10)²= 900 ${cm}^{\frac{2}{5}}$

5. A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

The area A of the circle with radius π is A = πr²

Then, rate of change in area $\frac{dA}{d t} = \frac{d \overset{?}{x} r^{2}}{d t}$ = $\frac{d x r^{2}}{d r} \cdot \frac{d r}{d t}$

= 2πr $dr$
${\frac{}{d t}}_{.}$

Q The wave moves at a rate 5cm/s we have,

$\frac{d r}{dF}$ = 5cm/s

So, $\frac{dA}{d t}$ =r. 5 = 10πr. ${cm}^{\frac{2}{5}}$

When r = 8 cm.

$\frac{d}{d t}$ = 10.π.8 cm ${cm}^{\frac{2}{5}}$ = 80 × cm ${cm}^{\frac{2}{5}}$

6. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

The circumference C of the circle with radius r is C = 2πr.

Then, rate of change in circumference is $\frac{dC}{d t} = \frac{d}{d t} 2 π r$ = $2 π \frac{d x}{d f} .$

Q Radius of circle increases at rate 0.7 cm/s we get,

$\frac{d y}{d t} = 0.7$ cm/s

So, $\frac{dC}{d t}$ = 2.× 0.7 cm/s = 1.4 × cm/s.

7. The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Since the length x is decreasing and the widthy is increasing with respect to time we have

$\frac{d x}{d t}$ = 5cm/min and $\frac{d y}{d t}$ = A cm/min

(a) The perimeter P of a rectangle will be, P = 2 (x + y)

Q Rate of change of perimeter, $\frac{d P}{d t} = \frac{d^{2}}{d t} (x + y)$

$= 2 (\frac{d x}{d t} + \frac{d y}{d t})$

= 2 ( -5 + 4)

= -2 cm/min

(b) The area A of the rectangle is A = x. y.

Rate of change of area is $\frac{dA}{d t} = \frac{d}{d t} (x \cdot y)$

$= x \frac{d y}{d t} + \frac{d x}{d t} \cdot y .$

= 4x - 5y

So, $\frac{dA}{d t}$ |x = 8ay = 6cm = 4 (8) - 5 (6) = 32 - 30 = 2 cm²/min spherical balloon

8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm

Let ‘r’ cm be the radius of volume V. measured Then,

$V = \frac{4}{3} π r^{3}$

Now, rate at which balloon is being inflated = 900 ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d v}{d r} = 900$ ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d t} (\frac{4}{3} π r^{3})$ = 900 ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d r} (\frac{4}{3} \times r^{3}) \frac{d r}{d t} = 900$

$\frac{4}{3}$ $π$ × 3 × r² $\frac{d r}{d r}$ = 900.

$\Rightarrow \frac{d r}{d t} = \frac{900}{4 π r^{2}} .$

When r = 15cm,

${\frac{d s}{d t} |}_{r = 15} = \frac{900}{4 π \times {(15)}^{2}}$ = $\frac{900}{900 π} = \frac{1}{π}$ cm/s.

Q Radius of balloon increases by $\frac{1}{π}$ per second.

9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm

The volume v of a spherical balloon with radius r is V. $\frac{4}{3} π r^{3} .$

with respect its radius.

Then, the rate of change of volume $| \frac{dV}{d r} = \frac{d}{d r} (\frac{4}{3} \cdot π r^{3})$

$= \frac{4}{3} π (\frac{d}{d r} r^{3})$

$= \frac{4}{3} π \times 3 \times π^{2}$

= 4 $π$ r²

Whenx = 10 cm,

$\frac{dV}{d r}$ = 4 $π$ 10)² = 400 $π$ cm³/cm

10. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

Since, the bottom of ground is increasing with time t,

$\frac{d x}{d t}$ = 2cm/s

From fig, Δ ABC, by Pythagorastheorem

AB² + BC² = AC²

$\Rightarrow$ x² + y² = 5²

x² + y² = 25 ____ (1)

Differentiating eqⁿ (1) w. r. t. time t we get,

$\frac{d}{d t} (x^{2} + y^{2}) = \frac{d}{d t} (25)$

$\Rightarrow 2 x \frac{d x}{d t} + 2 y \frac{d y}{d t} = 0 .$

$\Rightarrow 2 x \times 2 + 2 y \frac{2 y}{d y} = 0$

$\Rightarrow \frac{d y}{d t} = - \frac{2 x}{y}$ m/s

When x = 4m, the rate at which its height on the wall decreases is

$\frac{d y}{d t} = - \frac{2 \times 4}{3}$ ${\begin{cases} ∴ 4^{2} + y^{2} = 5^{2} \\ \Rightarrow y^{2} = 25 - 16 \\ \Rightarrow y √9 (L engthcan'tbenegetive) \\ \Rightarrow y = 3 \end{cases}$

$\Rightarrow \frac{d y}{d t} = - \frac{8}{3}$ room

12. The radius of an air bubble is increasing at the rate of 1 2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Let x be the radius of the bubble with volume .V. then,

$\frac{d r}{d t} = \frac{1}{2}$ cm/s

andV = $\frac{4}{3} π n^{3}$

Rate of change of volume $\frac{d V}{d t} = \frac{d}{d t} (\frac{4}{3} π^{3})$ = $\frac{4}{3} π \frac{d}{d t^{4}} r^{3}$

$= \frac{4}{3} π \times 3 r^{2} \frac{d r}{d t} .$

= 4πr² × $\frac{1}{2}$

= 2πr².

${\frac{d v}{d t} |}_{r = c m}$ = 2x (1)² 2π. ${cm}^{\frac{3}{5}}$

16. The total revenue in Rupees received from the sale of x units of a product is given by R (x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.

Given, R (x) = 13x² + 26x + 15.

Marginal revenue is the rate of change of total revenue with respect to the number of units sold Marginal revenue (MR) = $\frac{d R (x)}{d x}$

$= \frac{d}{d} (13 x^{2} + 26 x + 15)$

= 13 * 2x + 26

= 26x + 26

When x = 7,

MR = 26 * 7 + 26 = 182 + 26 = 208.

Hence, the required marginal reverse = ' 208.

Choose the correct answer for questions 17 and 18.

17. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π (B) 12π (C) 8π (D) 11π

The area A of the isle with radius r is given by with respect to radius r A = πr².

Then, rate of change of area of the circle $\frac{d \cdotA}{d x} = \frac{d π r^{2}}{d r}$

= 2πr.

When r = 6 cm

$\frac{dA}{d t} = 2 π \times 6 = 12 π .$

Q option (B) is correct.

39. Find the slope of the tangent to the curve $y = \frac{x - 1}{x - 2}, x \neq 2, a t, x = 10$

The given eqⁿ of the curve is $y = \frac{x - 1}{x - 2}$

Slope of tangent at x = 10 is given by,

${\frac{d y}{d x} |}_{x = 10} = {\frac{(x - 2) \frac{d}{d x} (x - 1) - (x - 1) \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}} |}_{x = 10}$

$= {\frac{(x - 2) - (x - 1)}{{(x - 2)}^{2}} |}_{x = 10} = {\frac{x - 2 - x + 1}{{(x - 2)}^{2}} |}_{x = 10}$

$= {\frac{- 1}{{(x - 2)}^{2}} |}_{x = 10}$

$= \frac{- 1}{{(10 - 2)}^{2}} = \frac{- 1}{8^{2}} = - \frac{1}{64}$

40. Find the slope of the tangent to curve y = x³ – x + 1 at the point whose x- coordinate is 2.

Slope of tangent to the given curve $y = x^{3} - x + 1$ is

$\frac{d y}{d x} = 3 x^{2} - 1 .$

So, ${\frac{d y}{d x} |}_{x = 2} = 3 {(2)}^{2} - 1 = 12 - 1 = 11 .$

41. Find the slope of the tangent to the curve y = x³ –3x + 2 at the point whose x-coordinate is 3.

Slope of tangent to the given curve $y = x^{3} - 3 x + 2$ is $\frac{d y}{d x} = 3 x^{2} - 3 .$

so, ${\frac{d y}{d x} |}_{x = 3} = 3 {(3)}^{2} - 3 = 27 - 3 = 24 .$

43. Find the slope of the normal to the curve $x = 1 - s i n θ, y = b c o s^{2} θ, a t, θ = \frac{π}{2}$

The given eqⁿ of the curves are

$x = 1 - a s i n θ y = b c o s^{2} θ$

so, $\frac{d x}{d θ} = - a c o s θ \frac{d y}{d θ} = - 2 b c o s θ s i n θ$

$∴ \frac{d y}{d x} = \frac{d y / d θ}{d x / d θ} = \frac{- 2 b c o s θ s i n θ}{a c o s θ} = \frac{2 b}{a} s i n θ$

Slope of tangent to curve at $θ = \frac{π}{2}$ is ${\frac{d y}{d x} |}_{θ = \frac{π}{2}}$

$= \frac{2 b}{a} s i n \frac{π}{2}$

$= \frac{2 b}{a}$

Hence, slope of normal to curve $= \frac{- 1}{2 b / a} = - \frac{a}{2 b}$

45. Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4)

Let the point joining the chord be $(2, 0) (4, 4)$

Then slope of the chord $= \frac{4 - 0}{4 - 2} {? Slope= \frac{y_{2} - y_{1}}{x_{2} - x_{1}}}$

$= \frac{4}{2}$

= 2

The given eqⁿ of the curve $y = {(x - 2)}^{2}$

slope of the tangent to the curve $\frac{d y}{d x} = 2 (x - 2) .$

Given that, the tangent is parallel to the chord PQ.

slope of tangent = slope of PQ.

$\Rightarrow 2 (x - 2) = 2 .$

$\Rightarrow x = 1 + 2$

$\Rightarrow x = 3 .$

and $y = {(3 - 2)}^{2} = 1^{2} = 1 .$

The required point on curve is $(3, 1)$

46. Find the point on the curve y = x³ – 11x + 5 at which the tangent is y = x – 11.

The given eqⁿ of the curve is $y = x^{3} - 11 x + 5$

slope of tangent to the curve $\frac{d y}{d x} = 3 x^{2} - 11$

Then eqⁿ of tangent is $y = x - 11$ $\Rightarrow x - y - 11 = 0$ which gives us slope $= \frac{- 1}{- 1} = 1$

So, $3 x^{2} - 11 = 1$

$\Rightarrow 3 x^{2} = 1 + 11 = 12$

$\Rightarrow x^{2} = 4$

$\Rightarrow x = \pm 2$

When x = 2, $y = 2^{3} - 11 (2) + 5 = 8 - 22 + 5 = - 9 .$

And when x = 2, $y = {(- 2)}^{3} - 11 (- 2) + 5 = - 8 + 22 + 5 = 19 .$

The point $(2, - 9)$ when put into $y = x - 11 .$ we get

$- 9 = 2 - 11$

$\Rightarrow - 9 = - 9$ which is true.

and the point $(- 2, 19)$ when put into $y = x - 11$ gives,

$19 = - 2 - 11$

$\Rightarrow 19 = - 13$ which is not true.

Hence, the required point is $(2, - 9)$

47. Find the equation of all lines having slope – 1 that are tangents to the curve $y = \frac{1}{x - 1}, x \neq 1$

The given eqⁿ of curve is $y = \frac{1}{x - 1}$

Slope of tangent to the given curve is $\frac{d y}{d x} = \frac{- 1}{{(x - 1)}^{2}}$

Given that, slope of tangent = 1.

$\to \frac{- 1}{{(x - 1)}^{2}} = - 1 .$

$\Rightarrow {(x - 1)}^{2} = 1 .$

$\Rightarrow x - 1 = \pm 1$

$\to x = 1 \pm 1 .$

ie, X=1+1 or $x = 1 - 1$

$\Rightarrow x = 2$ or $x = 0$

When $x = 2, y = \frac{1}{2 - 1} = 1$

and when $x = 0, y = \frac{1}{0 - 1} = - 1 .$

Hence, the point of contact of the tangents are $(2, 1)and (0, - 1)$

The reqd. eqⁿ of line are $y - 1 = (- 1) (x - 2) {\begin{matrix} ? y - y_{0} = m (x - x_{0}) & eqn^{} of line \end{matrix}$

and $y - (- 1) = (- 1) (x - 0) .$

$\Rightarrow y - 1 = - x + 2$ and $y + 1 = - x$

$\to x + y - 3 = 0$ and $x + y + 1 = 0 .$

48. Find the equation of all lines having slope 2 which are tangents to the curve

$y = \frac{1}{x - 3}, x \neq 3$

The given eqⁿ of curve is $y = \frac{1}{x - 3}$

Slope of tangent to the curve is $\frac{d y}{d x} = \frac{- 1}{{(x - 3)}^{2}} .$

Given, $\frac{d y}{d x} = 2$

$\Rightarrow \frac{- 1}{{(x - 3)}^{2}} = 2$

$\Rightarrow {(x - 3)}^{2} = - \frac{1}{2}$ which is not possible

we conclude that there is no possible tangent to the given curve with slope = 2.

49. Find the equations of all lines having slope 0 which are tangent to the curve

$y = \frac{1}{x^{2} - 2 x + 3}$

The given eqⁿ of the curve is $y = \frac{1}{x^{2} - 2 x + 3}$

Slope of tangent to the curve is $\frac{d y}{d x} = \frac{- 1}{(x^{2} - 2 x + 3)} = \frac{d}{d x} (x^{2} - 2 x + 3)$

$= \frac{- (2 x_{1} - 2)}{{(x^{2} - 2 x + 3)}^{2}}$

Given, $\frac{d y}{d x} = 0$

$\to \frac{- (2 x - 2)}{{(x^{2} - 2 x + 3)}^{2}} = 0$

$\Rightarrow - 2 (x - 1) = 0$

$\Rightarrow x = 1$

When $x = 1, y = \frac{1}{1^{2} - 2 \times 1 + 3} = \frac{1}{1 - 2 + 3} = \frac{1}{2}$

The point of contact of the tangent to the curve is $(1, \frac{1}{2})$

The eqⁿ of the line is $y - \frac{1}{2} = 0 (x - 1)$

$\to y = \frac{1}{2}$

50. Find points on the curve $\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1$ at which the tangents are

(i) Parallel to x-axis (ii) Parallel to y-axis.

Diffrentiating $\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1 .$ wrt. X we get,

$\frac{2 x}{9} + \frac{2 y}{16} \frac{d y}{d x} = 0$

$\Rightarrow \frac{2 y d y}{16 d x} = - \frac{2 x}{9}$

$\Rightarrow \frac{d y}{d x} = \frac{- 16 x}{9 y}$

(i) When the tangent is || to x-axis, the slope of tangent is 0

ie, $\frac{d y}{d x} = 0$

$\Rightarrow \frac{- 16 x}{9 y} = 0$

$\Rightarrow x = 0$ putting this in the eqⁿ of curve. We get,

$\to \frac{0^{2}}{9} + \frac{y^{2}}{16} = 1$

$\Rightarrow y^{2} = 16$

$\Rightarrow y = \pm 4 .$

The point at which the tangents are parallel to x-axis are $(0, 4)and (0, - 4)$

(ii) When the tangent is parallel to y-axis, the slope of the normal is 0.

ie, $\frac{- 1}{\frac{d y}{d x}} = 0$

$\Rightarrow - \frac{d x}{d y} = 0$

$\Rightarrow \frac{9 y}{16 x} = 0$

$\to y = 0$ , putting this in the eqⁿ of curve we get,

$\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1 .$

$\to x^{2} = 9$

$\to x = \pm 3$

The point at which the tangents are parallel to y-axis are $(3, 0)and (- 3, 0)$

51. Find the equations of the tangent and normal to the given curves at the indicated points:

$\begin{array}{l} (i) y = x^{4} - 6 x^{3} + 13 x^{2} - 10 x + 5, a t, (0, 5) \\ (i i) y = x^{4} - 6 x^{3} + 13 x^{2} - 10 x + 5, a t, (1, 3) \\ (i i i) y = x^{3}, a t, (1, 1) \\ (i v) y = x^{2}, a t, (0, 0) \\ (v) x = c o s y, y = s i n t, a t, t = \frac{π}{4} \end{array}$

(i) we have, $y = x^{4} - 6 x^{3} + 13 x^{2} - 10 x + 5$

slope of tangent, $\frac{d y}{d x} = 4 x^{3} - 18 x^{2} + 26 x - 10$

${\frac{d y}{d x} |}_{(x, y) = (0, 5)} = - 10 .$

slope of normal $= \frac{- 1}{- 10} = \frac{1}{10}$

Hence eqⁿ of tangent at (0, 5) is

$y - 5 = - 10 (x - 0) \to 10 x + y - 5 = 0$

And eqⁿ of normal at (0, 5) is

$y - 5 = \frac{1}{10} (x - 0)$

$\Rightarrow 10 y - 50 = x$

$\Rightarrow x - 10 y + 50 = 0$

(ii) We have, $y = x^{4} - 6 x^{3} + 13 x^{2} - 10 x + 5$

Slope of tangent, $\frac{d y}{d x} = 4 x^{3} - 18 x^{2} + 26 x - 10 .$

${\frac{d y}{d x} |}_{(x, y) = (1, 3)} = 4 {(1)}^{3} - 18 {(1)}^{2} + 26 (1) - 10$

$= 4 - 18 + 26 - 10$

= 30 28

= 2

Slope of normal $= - \frac{1}{2}$

Hence eqⁿ of tangent at (1, 3) is

$y - 3 = 2 (x - 1)$

$\Rightarrow y - 3 = 2 x - 2$

$\to 2 x - y + 1 = 0$

And eqⁿ of normal at (1,3) is

$(y - 3) = - \frac{1}{2} (x - 1)$

$\Rightarrow 2 y - 6 = - x + 1$

$\to x + 2 y - 7 = 0$

(iii) We have, $y = x^{3}$

Slope of tangent, $\frac{d y}{d x} = 3 x^{2}$

${\frac{d y}{d x} |}_{(1, 1)} = 3 {(1)}^{2} = 3 .$

And slope of normal $= - \frac{1}{3}$

Hence, eqn of tangent at (1, 1) is

$y - 1 = 3 (x - 1)$

$\Rightarrow y - 1 = 3 x - 3$

$\to 3 x - y - 2 = 0$

And eqⁿ of normal at (1,1) is

$y - 1 = - \frac{1}{3} (x - 1)$

$\to 3 y - 3 = - x + 1$

$\Rightarrow x + 3 y - 4 = 0 .$

(iv) We have, $y = x^{2}$

Slope of tangent $\frac{d y}{d x} = 2 x$

${\frac{d y}{d x} |}_{(0, 0)} = 0 .$

So, eqⁿ of the tangent at (0,0) is

$(y - 0) = 0 (x - 0)$

$\Rightarrow y = 0 .$

ie, x- axis

Hence, the eqⁿ of normal is x = 0 ie, y-axis

(v) We have, $x = c o s t y = s i n t$

$\frac{d x}{d t} = - s i n t \cdot \frac{d y}{d t} = c o s t$

So, slope of tangent $\frac{d y}{d x} = \frac{d y / d t}{d x / d t} = \frac{c o s t}{- s i n t} = - c o t t$

${\frac{d y}{d x} |}_{t = \frac{π}{4}} = - c o t \frac{π}{4} = - 1 .$

And slope of normal $= \frac{- 1}{- 1} = 1$

52. Find the equation of the tangent line to the curve y = x² – 2x +7 which is

(a) Parallel to the line 2x – y + 9 = 0 (b) Perpendicular to the line 5y – 15x = 13

The eqⁿof the given curve is $y = x^{2} - 2 x + 7$

Slope of tangent, $\frac{d y}{d x} = 2 x - 2$

(a) The line $2 x - y + 9 = 0 \Rightarrow y = 2 x + 9$ compared to $y = m x + c$ gives,

Slope of line = 2.

If the tangent of the curve is parallel to the line

$\frac{d y}{d x} =slope of line$

$\to 2 x - 2 = 2$

$\to x = \frac{4}{2} \Rightarrow x = 2$

When $x = 2, y = {(2)}^{2} - 2 (2) + 7 = 7$

Hence, the point of contact of the tangent is (2, 7)

The eqⁿ of tangent is $y - 7 = 2 (x - 2)$

$\Rightarrow y - 7 = 2 x - 4$

$\to 2 x - y + 3 = 0$

(b) The line $5 y - 15 x = 13 \to y = \frac{15}{5} x + \frac{13}{5} \Rightarrow y = 3 x + \frac{13}{5}$

compared to $y = m x + c$ gives

slope of line = 3

As the tangent to the curve is ⊥ to the line.

$\frac{d y}{d x} = -1/slope opf line$

$\Rightarrow 2 x - 2 = - \frac{1}{3}$

$\Rightarrow 6 x - 6 = - 1$

$\to 6 x = 5$

$\to x = \frac{5}{6}$

When $x = \frac{5}{6}$ we get $y = {(\frac{5}{6})}^{2} - 2 \times \frac{5}{6} + 7$

$= \frac{25}{36} - \frac{5}{3} + 7$

$= \frac{25 - 60 + 252}{36} = \frac{217}{36}$

Hence, the point of contact of the tangent is $(\frac{5}{6}, \frac{217}{36})$

And eqⁿ of the tangent is

$y - \frac{217}{36} = \frac{- 1}{3} (x - \frac{5}{6})$

$\Rightarrow 3 y - \frac{217}{12} = - x + \frac{5}{6}$

$\Rightarrow x + 3 y - \frac{217}{12} - \frac{5}{6} = 0$

$\Rightarrow x + 3 y - \frac{217 - 10}{12} = 0$

$\Rightarrow x + 3 y - \frac{227}{12} = 0$

$\Rightarrow 12 x + 36 y - 227 = 0 .$

53. Show that the tangents to the curve y = 7x³ + 11 at the points where x = 2 and x = – 2 are parallel.

The given eqⁿ of the curve is $y = 7 x^{3} + 11$ .

Slope of tangent $\frac{d y}{d x} = 21 x^{2}$

${\frac{d y}{d x} |}_{x = 2} = 21 {(2)}^{2} = 21 \times 4 = 84 .$

and ${\frac{d y}{d x} |}_{x = - 2} = 21 {(- 2)}^{2} = 21 \times 4 = 84$

The tangent to the given curve at x = 2 and x = -2 are parallel.

54. Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point.

The given eqⁿ of the curve is $y = x^{3}$ .

Slope of tangent, $\frac{d y}{d x} = 3 x^{2}$

As, slope of tangent = y – coordinate of the point.

$\frac{d y}{d x} = y$

$\Rightarrow 3 x^{2} = x^{3}$

$\to 3 x^{2} - x^{3} = 0$

$\Rightarrow x^{2} (3 - x) = 0$

$\Rightarrow x = 0 x = 3$

When $x = 0, y = 0$

and when $x = 3 \cdot, y = 3^{3} = 27 .$

The required points are $(0, 0)and (3, 27)$

55. For the curve y = 4x³ – 2x⁵ , find all the points at which the tangent passes through the origin.

The given eqⁿ of the curve is $y = 4 x^{3} - 2 x^{5}$

Slope of tangent, $\frac{d y}{d x} = 12 x^{2} - 10 x^{4} .$ ________(1)

Let P(x, y) be the required point at the tangent passing through the origin (0,0)

Then, $\frac{d y}{d x} = \frac{y - 0}{x - 0} = \frac{y}{x}$ _________(2)

So, from (1) and (2) we get,

$\frac{y}{x} = 12 x^{2} - 10 x^{4} .$

$\Rightarrow y = 12 x^{3} - 10 x^{5} .$

Putting this value of y in the eqⁿ of curve we get,

$12 x^{3} - 10 x^{5} = 4 x^{3} - 2 x^{5} .$

$\to 8 x^{3} - 8 x^{5} = 0$

$\Rightarrow 8 x^{3} (1 - x^{2}) = 0$

$\Rightarrow x^{3} = 0$ or $1 - x^{2} = 0$

$\Rightarrow x = 0$ or $x = \pm 1 .$

When, $x = 0, y = 4 {(0)}^{3} - 2 {(0)}^{5} = 0$

$x = 1, y = 4 {(1)}^{3} - 2 {(1)}^{5} = 4 - 2 = 2$

$x = - 1, y = 4 {(- 1)}^{3} - 2 {(- 1)}^{5} = - 4 + 2 = - 2$

The required points are (0,0), (1,2) and (1,2)

56. Find the points on the curve x² + y² – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

The given eqⁿof the curve is $x^{2} + y^{2} - 2 x - 3 = 0$ ________ (1)

Differentiating the given curve wrt.x we get,

$2 x + 2 y \frac{d y}{d x} - 2 = 0 .$

$\Rightarrow 2 y \frac{d y}{d x} = 2 - 2 x$

$\to \frac{d y}{d x} = \frac{1 - x}{y}$ slope of tangent

Given, tangent is | to x-axis

ie, $\frac{d y}{d x} = 0$

$\Rightarrow \frac{1 - x}{y} = 0$

$\Rightarrow x = 1$

Putting x = 1 in eqⁿ (1) we get,

$1^{2} + y^{2} - 2 (1_{i}) - 3 = 0$

$\Rightarrow y^{2} - 4 = 0$

$\Rightarrow y^{2} = 4$

$\Rightarrow y = \pm 2$

Hence, the required points are (1,2) and (1, 2).

57. Find the equation of the normal at the point (am² ,am³ ) for the curve ay² = x³

The given eqⁿ of curve is $a y^{2} = x^{3}$ _____(1)

Differentiating eqⁿ (1) wrt.x we get,

$2 a y \frac{d y}{d x} = 3 x^{2} .$

$\Rightarrow \frac{d y}{d x} = \frac{3 x^{2}}{2 a y}$ , slope of tangent

${\frac{d y}{d x} |}_{(x, y) = (a m^{2}, a m^{3})} = \frac{3 {[a m^{2}]}^{2}}{2 a [a m^{3}]} = \frac{3 a^{2} m^{4}}{2 a^{2} m^{3}} = \frac{3 m}{2}$

corresponding slope of normal $= \frac{- 1}{(3 m / 2)} = - \frac{2}{3 m}$

Hence, eqⁿ of normal at $(a m^{2}, a m^{3})$ is

$y - a m^{3} = - \frac{2}{3 m} (x - a m^{2}) .$

$\Rightarrow 3 m y - 3 a m^{4} = - 2 x + 2 a m^{2}$

$\Rightarrow 2 x + 3 m y - 3 a m^{4} - 2 a m^{2} = 0$

$\Rightarrow 2 x + 3 m y - a m^{2} (2 + 3 m^{2}) = 0$

58. Find the equation of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

The given eqⁿ of the curve is $y = x^{3} + 2 x + 6$ _____(1)

slope of tangent to the curve, $\frac{d y}{d x} = 3 x^{2} + 2$

so, slope of normal to the curve $= \frac{- 1}{3 x^{2} + 2}$

Now, the line $x + 14 y + 4 = 0 \to y = - \frac{1}{14} x - \frac{4}{14}$ compared to $y = m x + c$ gives

slope of line = $- \frac{1}{14}$

As the normal is parallel to the line

$\Rightarrow \frac{- 1}{3 x^{2} + 2} = - \frac{1}{14}$

$\to 3 x^{2} + 2 = 14$

$\Rightarrow 3 x^{2} = 12$

$\to x^{2} = 4$

$\Rightarrow x = \pm 2$

When $x = 2, y = {(2)}^{3} + 2 (2) + 6 = 8 + 4 + 6 = 18$

and when $x = - 2, y = {(- 2)}^{3} + 2 (- 2) + 6 = - 8 - 4 + 6 = - 6$

The point of contact of the normal are (2, 18) and (-2, -6)

Hence the eqⁿ of normal are

$y - 18 = - \frac{1}{14} (x - 2)$ and $y - (- 6) = \frac{- 1}{14} [x - (- 2)]$

$\Rightarrow 14 y - 252 = - x + 2$ $\Rightarrow y + 6 = \frac{- 1}{14} (x + 2)$

$\Rightarrow x + 14 y - 254 = 0$ $\Rightarrow 14 y + 84 = - x - 2$

$\Rightarrow x + 14 y + 86 = 0 .$

59. Find the equations of the tangent and normal to the parabola y² = 4ax at the point (at² , 2at).

The given of the parabola is $y^{2} = 4 a x$

slope of tangent is given by

$\frac{d}{d x} y^{2} = \frac{d}{d x} 4 a x$

$\to 2 y \frac{d y}{d x} = 4 a$

$\Rightarrow \frac{d y}{d x} = \frac{2 a}{y} .$

${\frac{d y}{d x} |}_{(a t^{2}, 2 a t)} = \frac{2 a}{2 a t} = \frac{1}{t}$

so, slope of normal

so, slope of normal $= \frac{- 1}{(\frac{1}{t})} = - t$

Hence eqⁿ of tangent at point $(a t^{2}, 2 a t)$ is

$y - 2 a t = \frac{1}{t} (x - a t^{2})$

$\to t y - 2 a t^{2} = x - a t^{2}$

$\to x - t y - a t^{2} + 2 a t^{2} = 0$

$x - t y + a t^{2} = 0$

And eqⁿ of normal at point $(a t^{2}, 2 a t)$ is

$(y - 2 a t) = (- t) (x - a t^{2})$

$\to y - 2 a t = - t x + a t^{3}$

$\Rightarrow t x + y - 2 a t + a t^{3} = 0$

$\Rightarrow t x + y - a t (2 + t^{2}) = 0$

60. Prove that the curves x = y²and xy = k cut at right angles* if 8k² = 1.

The given eqⁿ of the curves are

$x = y^{2}$ ________(1)

and $x y = x$ ___________(2)

Differentiating eqn (1) and (2) wrt ‘x’ we get,

$\frac{d y^{2}}{d x} = \frac{d x}{d x}$

$\to 2 y \frac{d y}{d x} = 1$

$\to \frac{d y}{d x} = \frac{1}{2 y}$ _________(3)

and $\frac{d}{d x} (x y) = \frac{d}{d x} (x)$

$\to x \frac{d y}{d x} + y = 0$

$\to \frac{d y}{d x} = - \frac{y}{x}$ _________(4)

Since the two curves cut each other at right angles we get,

$(\frac{1}{2 y}) \times (\frac{- y}{x}) = - 1 .$

$\to 2 x = 1$ ________(5)

The point of intersection can be solve from eqn (1) and (2),

$y^{2} . y = k$

$\to y^{3} = k$

$\to y = k^{1 / 3}$

$∴ x = y^{2} = r^{2 / 3} .$

Hence, using eqⁿ (5) we get,

$2 {(r)}^{2 / 3} = 1 .$

$\Rightarrow {[2 r^{2 / 3}]}^{3} = 1^{3}$

$\Rightarrow 8 r^{2} = 1$

Hence proved

61. Find the equations of the tangent and normal to the hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at the point $(x_{0}, y_{0})$

The given eqⁿof the hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ______(1)

Differentiating eqn (1) wrt ‘x’ we get,

$\frac{2 x}{a^{2}} - \frac{2 y}{b^{2}} \frac{d y}{d x} = 0$

$\Rightarrow \frac{2 y}{b^{2}} \frac{d y}{d x} = \frac{2 x}{a^{2}}$

$\Rightarrow \frac{d y}{d x} = \frac{b^{2} x}{a^{2} y}$

${\frac{d y}{d x} |}_{(x, y)} = (x_{0}, y_{0}) \frac{b^{2} x_{0}}{a^{2} y_{0}}$ is the reqd slope of tangent to the curve

So, eqⁿof tangent at point $(x_{0}, y_{0})$ is

$y - y_{0} = \frac{b^{2} x_{0}}{a^{2} y_{0}} (x - x_{0}) .$

$\Rightarrow \frac{y y_{0}}{b^{2}} - \frac{{y_{0}}^{2}}{b^{2}} = \frac{x x_{0}}{a^{2}} - \frac{{x_{0}}^{2}}{a^{2}}$

$\Rightarrow \frac{x x_{0}}{a^{2}} - \frac{y y_{0}}{b^{2}} = \frac{{x_{0}}^{2}}{a^{2}} - \frac{{y_{0}}^{2}}{b^{2}}$

As $(x_{0}, y_{0})$ lies on the parabola given by eqn (1) we write,

$\frac{{x_{0}}^{2}}{a^{2}} - \frac{{y_{0}}^{2}}{b^{2}} = 1$

Hence, $\frac{x x_{0}}{a^{2}} - \frac{y y_{0}}{b^{2}} = 1$

64. The line y = x + 1 is a tangent to the curve y² = 4x at the point (A) (1, 2) (B) (2, 1) (C) (1, – 2) (D) (– 1, 2)

The given eqn of curve is $y^{2} = 4 x$ .

Then, differentiating wrt x we get,

$2 y \frac{d y}{d x} = 4$

$\Rightarrow \frac{d y}{d x} = \frac{2}{y}$ which is the slope of the tangent to the curve.

The line $y = x + 1$ compared to $y = m x + c$ gives slope of line = 1.

Since, tangent is the line we have,

$\frac{2}{y} = 1 .$

$\Rightarrow y = 2$

Putting y = 2 in $y^{2} = 4 x$ we get, $4 x = 2^{2} \Rightarrow x = 1 .$

Hence, the required point is $(1, 2)$

Option (A) is correct.

65. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) Let y= ?x : Let x = 25 and x = 0. 3.

Then, ?y = ?x+?x $??x$

$=?25.3 ??25$

$=?25.3 ? 5$

$??25.3 = 5 + ? y ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? y ~ d y .$

= 5 + dy

= $5 + (\frac{d y}{d x}) A x .$

$= 5 + \frac{1}{2?x} ? x .$

$= 5 + \frac{1}{2?25} ? 0.3$

$= 5 + \frac{0.3}{2 \times 5}$

= 5 + 0.03

$??25.3 = 5.030$

(ii) ?49.5

A.(ii)

Let y = ?x $.$ Let x = 49 and x = 0.5.

Then, $? y =?x+?x ??x .$

$=?49.5 ??49$

$??49.5 = 7 + ? y = 7 + {(\frac{d y}{d x})}_{}^{? x}$

$= 7 + \frac{1}{2?x} ? x$

$= 7 + \frac{1}{2?49} \times (0.5)$

$= 7 + \frac{0.5}{14}$

= 7 + 0.0357.

$??49.5 = 7 ? 035$

(iii) ?0.6

A.(iii)

Let y = ?x $.$ Let x = 0.64 and ?x = 0.04.

Then, $? y =?x+Ax ??x$

$=?0.64-0.04 ??0.64 .$

$=?0.6 ? 0.8$

$??0.6= 0.8 + ? y = 0.8 + (\frac{d y}{d x}) ? x$

$= 0.8 + \frac{1}{2?x} ? x$

$= 0.8 + \frac{1}{2?0.64} \times (? ? 0.04)$

$= 0.8 \frac{? 0.04}{2 \times 0.8}$

= 0.8 - 0.025.

= 0.775.

(iv) ${(0.009)}^{\frac{1}{3}}$

A.(iv)

Let $y = x^{\frac{1}{3}}$ Let x = 0.008 and ?x = 0.00 1.

Then, ?y = ${[x + ? x]}^{\frac{1}{3}} ? {[x]}^{\frac{1}{3}}$

$= {(0.009)}^{\frac{1}{3}} ? {(0.008)}^{\frac{1}{3}}$

${(0.009)}^{\frac{1}{3}} = 0.2 + ? y = 0.2 + \frac{d y}{d x} ? ? x$

$= 0.2 + \frac{1}{3 {(x^{\frac{1}{3}})}^{2}} ? x$

$= 0.2 + \frac{1}{3 {[{(0.008)}^{\frac{1}{3}}]}^{2}} \times (0.001)$

$= 0.2 + \frac{0.001}{3 \times {(0.0)}^{2}} = 0.2 + \frac{0.001}{0.12}$

= 0.2 + 0.0083.

= 0.208.

(v) ${(0.999)}^{\frac{1}{10}}$

A.(v)

Let $y = x^{\frac{1}{10}} .$ Let x = 1 and ?x = -0.001

Then, $? y = {(x + ? x)}^{\frac{1}{10}} ? x^{\frac{1}{10}}$

$= {(0.999)}^{\frac{1}{10}} ? {(1)}^{\frac{1}{10}}$

$? 1 {(0.999)}^{\frac{1}{10}} = 1 + A y$

$= 1 + \frac{d y}{d x} ? x .$

$= 1 + \frac{1}{10 x^{\frac{9}{10}}} \times ? x$

$= 1 ? \frac{0.001}{10 (. 1^{\frac{9}{10}})} = 1 ? \frac{0.001}{10} = 1 ? 0.0001$

= 0.999.

(vi) ${(15)}^{\frac{1}{4}}$

A.(vi)

Let $y = x^{\frac{1}{4}} .$ Then, x = 16 and ?x = 1.

Then, $? y = {(x + A x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(15)}^{\frac{1}{4}} ? {(16)}^{\frac{1}{4}}$

$? {(15)}^{\frac{1}{4}} = 2 + A y = 2 + \frac{d y}{d x} A x$

$= 2 + \frac{1}{4 x^{\frac{3}{4}}} ? (? 1)$

$= 4 ? \frac{1}{4 {(16)}^{\frac{3}{4}}}$

$= 2 ? \frac{1}{4 \times 8}$

$= 4 ? \frac{1}{32}$

$= \frac{64 ? 1}{32} = \frac{63}{32} = 1.968$

(vii) ${(26)}^{\frac{1}{3}}$

A.(vii)

Let $y = x^{\frac{1}{3}} .$ Let x = 27 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{3}} ? x^{\frac{1}{3}}$

$= {(26)}^{\frac{1}{3}} ? {(27)}^{\frac{1}{3}}$

$? {(26)}^{\frac{1}{3}} = 3 + 4 y = 3 + \frac{d y}{d x} A x = 3 + \frac{1}{3 x^{\frac{2}{3}}} A x,$

$= 3 ? \frac{1}{3 {(2 F)}^{\frac{2}{3}}}$

$= 3 ? \frac{1}{27}$

$= \frac{81 ? 1}{27} = \frac{80}{27} = 2.962 .$

(viii) ${(255)}^{\frac{1}{4}}$

A.(viii)

Let $y = x^{\frac{1}{4}} .$ Let x = 256 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{4}} ? x^{\frac{1}{4}} .$

$= {(255)}^{\frac{1}{4}} ? {(256)}^{\frac{1}{4}}$

$? {(255)}^{\frac{1}{4}} = {(256)}^{\frac{1}{4}} + 4 y = 4 + \frac{d y}{d x} A x .$

$= 4 + \frac{1}{4 x^{\frac{3}{4}}} ? 4 x$

$= 4 \frac{? 1}{4 {(256)}^{\frac{3}{4}}}$

$= 4 ? \frac{1}{256}$

$= \frac{1024 ? 1}{256}$

$= \frac{1023}{256} = 3.996$

(ix) ${(82)}^{\frac{1}{4}}$

A.(ix)

Let $y = x^{\frac{1}{4}} .$ let x = 81 and ?x = 1.

Then, $? y = {(x + 4 x)}^{\frac{1}{4}} ? x^{\frac{1}{4}}$

$= {(82)}^{\frac{1}{4}} ? {(81)}^{\frac{1}{4}}$

$7 {(82)}^{\frac{1}{4}} = 3 + 4 y = 3 + \frac{d y}{d x} ? x$

$= 3 + \frac{1}{4 x^{\frac{3}{4}}} ? ? x$

$= 3 + \frac{1}{4 {(81)}^{\frac{3}{4}}}$

$= 3 + \frac{1}{4 \times 3^{3}}$

$= 3 + \frac{1}{108} = \frac{324 + 1}{108} = \frac{325}{108} = 3.001 .$

(x) ${(401)}^{\frac{1}{2}}$

A.(x)

Let y = $x^{\frac{1}{2}}$ = ?x $.$ Let x = 400 and ?x = 1.

Then, $? y = {(x + ? x)}^{\frac{1}{2}} ? x^{\frac{1}{2}} .$

$? ? ? y = {(401)}^{\frac{1}{2}} ? {(400)}^{\frac{1}{2}}$

$? ? {(401)}^{\frac{1}{2}} = 20 ? + ? y = 20 + \frac{d y}{d x} ? x$

$= 20 + \frac{1}{2?x} ? x$

$= 20 + \frac{1}{2?400}$

$= 20 + \frac{1}{2 \times 20}$

$= 20 + \frac{1}{40}$

$= \frac{800 + 1}{40} = \frac{801}{40} = 20.025 .$

(xi) ${(0.0037)}^{\frac{1}{2}}$

A.(xi)

Let $y = x^{\frac{1}{2}}$ Let x = 0.0036 and ?x = 0.0001

Then, $? y =?x+?x ??x$

$=?0.0037 ??0.0036$

$??0.0037 = 0.06 + D y = 0.06 + \frac{d y}{d x} ? x = 0.06 + \frac{x}{2?x}$

$= 0.06 + \frac{0.001}{2?0.0036}$

$= 0.06 + \frac{0.001}{2 \times 0.06}$

$= 0.06 + \frac{0.001}{0.12 .}$

= 0.06 + 0.0008.

= 0.0608.

(xii) ${(26.57)}^{\frac{1}{3}}$

A.(xii)

Let $y = x^{\frac{1}{3}} .$ Let x = 27 and ?x = 0.43.

Then, $? y = {(x + A x)}^{\frac{1}{3}} ? {(x)}^{\frac{1}{3}}$

$= {(26.57)}^{\frac{1}{3}} ? {(27)}^{\frac{1}{3}}$

$? {(26.57)}^{\frac{1}{3}} = 3 + 4 y = 3 + \frac{d y}{d x} ? x = 3 + \frac{1}{3 x^{\frac{2}{3}}} ? ? x .$

$= 3 + \frac{1}{3 {(27)}^{\frac{2}{3}}} ? Q: 67. Find the approximate value of f (5.001), where f(x) = x 3 - 7x2 + 15.A: Given, y = f (x) = x 3 - 7 x 2 + 15.So, \frac{d y}{d x} = f^{'} (x) = 3 x^{2} - 14 x . \Rightarrow dy = (3 x 2 - 14 x) dx. \Rightarrow Δy = (3 x 2 - 14 x) Δx. Let, x = 5 and Δx = 0.001. Then, Δy = f (x + Δx) f (x). \Rightarrow f (x + Δx) = f (x) + Δy = f (x) + (3 x 2 - 4 x) Δx. \Rightarrow f (5 + 0.001) = 5 3 - 7 (5) 2 + 15 + [3 (5) 2 - 14 (5)]. (0.001). \Rightarrow f (5.001) = 125 - 175 + 15 + (75 - 70) (0.001) = -35 + 0.005 = - 34.995.Q: 69. Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% Read moreA: We know that, the surface area 5 of a ‘x’ when length cube, is S = 6 x 2 . So, d S = \frac{d S}{d x} Δ x = 12 x \cdot Δ x . Given decrease in side, Δ x = - 1 % x = - \frac{x}{100} . ∴ d S = 12 x (- \frac{x}{100}) = - 0.12 x^{2} m^{2} .Q: 70. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume. Read moreA: Let x be the radius of the sphere & Δπ be the error in measuring the radius. Then, π = 7m and Δr = 0.02m. Now, volume v of sphere is V = \frac{4}{3} π r^{3} . So, \frac{dU}{d x} = 4 π r^{2} \Rightarrow d v = (\frac{d v}{d r}) \cdot Δ r = 4 π r^{2} (Δ π) \Rightarrow dv = 4π (7) 2 (.0.02) = 3.92 πm3 ∴The appropriate error is calculating the volume is 3.92π m 3 .Q: 71. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area. Read moreA: Let be the radius of the sphere &r be the error in measuring the radius. Then, π = 9m and Δr = 0.03m. Now, surface area S of the sphere is S = 4π r 2 So, \frac{d s}{d r} = 8 π r . ∴e, this = (\frac{d s}{d r}) Δr = 8πr.Δr = 8π \times 9 \times 0.03 = 2.16π m 3 . Appropriate error in calculating the surface area is 2.16π m 3 .Q: 72. If f(x) = 3x 2 + 15x + 5, then the approximate value of f (3.02) is (A) 47.66 (B) 57.66 (C) 67.66 (D) 77.66 Read moreA: We have, y = f (x) = 3x 2 + 15x + 5. \frac{d y}{d x} = 6 x + 15 \Rightarrow dy = (6x + 15) dx \Rightarrow Δy = (6x + 15) Δx. Let, x = 3 and Δx = 0.02 then, Δy = f (x + Δx) - f (x) \Rightarrow f (x + Δx) = f (x) + Δy = f (x) + (6x + 5) Δx. \Rightarrow f (3 + 0.02) = 3 (3) 2 + 15 (3) + 5 + (6 \times 3 + 15) (0.02). \Rightarrow f (3.02) = 27 + 45 + 5 + (18 + 15) (0.02). = 77 + 0.66 = 77.66 ∴ Option (D) is correct.Q: 73. The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is (A) 0.06 x 3 m 3 (B) 0.6 x 3 m 3 (C) 0.09 x 3 m 3 (D) 0.9 x 3 m 3 Read moreA: The volume v of a cube with side ‘x’ metre is v = x 3 So, d v = (\frac{d v}{d x}) Δ x = 3 x^{2} Δ x . ∴increase in side, Δx = 3% of = \frac{3 x}{100} . ∴dv = 3 x 2 π \frac{3 x}{100} - 0.09 x^{3} m^{3} . Hence, option (C) is correct.Q: 75. Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = |x + 2| - 1 Read moreA: (i) we have, f(x) = |x + 2| - 1 We know that, for all x \in ?, | x + 2 | \geq 0 \Rightarrow | x + 2 | - 1 \geq - 1 . \Rightarrow f(x) \geq - 1. ∴ Minimum value of f(x) = -1 when x + 2 = 0 \Rightarrow x = - 2. And maximum value of f(x) does not exist. (ii) g (x) = - | x + 1 | + 3 A(ii) We have, g (x) = - | x + 1 | + 3 For all x \in, | x + 1 | \geq 0 . \Rightarrow - | x + 1 | \leq 0 \Rightarrow - | x + 1 | + 3 \leq 0 + 3 \Rightarrow g(x) \leq 3. ∴ Maximum value of g(x) = 3 when | x + 1 | = 0 \Rightarrow x = - 1 . And minimum value does not exist. (iii) h(x) = sin (2x) + 5. A(iii) we have, h(x) = sin (2x) + 5. For all x \in ?, - 1 \leq s i n 2 x \leq 1 . {range of sine function is [-1, 1]} \Rightarrow -1 + 5 \leq sin 2x + 5 \leq 1 + 5. \Rightarrow 4 \leq h(x) \leq 6. ∴ Maximum value of h(x) = 6. Minimum value of h(x) = 4. (iv) f (x) = | s i n 4 x + 3 | . A(iv) we have, f (x) = | s i n 4 x + 3 | . As for all x \in ?, - 1 \leq s i n 4 x \leq 1 \Rightarrow -1 + 3 \leq sin 4x + 3 \leq 1 + 3 \Rightarrow | 2 | \leq | s i n 4 x + 3 | \leq | 4 | . \Rightarrow 2 \leq f(x) \leq 4. ∴ Maximum value of f(x) = 4. Minimum value of f(x) = 2. (v) h(x) = x + 1.,x\in ( -1, 1). A(v) we have, h(x) = x + 1.,x\in ( -1, 1). Given, -1 - 1 + 1 < x + 1 < 1 + 1 0 < h (x) < 2. ∴ Maximum value and minimum value of h(x) does not exist.Q: 76. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) f(x) = x 2 . Read moreA: (i) we have, f(x) = x 2 . f(x) = 2x. andf(x) = 2. AR extreme point, f(x) = 0 2x = 0 x = 0. When x = 0, f(0) = 2 > 0. ∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0 2 = 0. (ii) g(x) = x 3 3x A(ii) we have, g(x) = x 3 - 3 x \Rightarrow g'(x) = 3 x 2 - 3 \Rightarrow g''(x) = 6x. At extreme point, g'(x) = 0 3 x 2 - 3 = 0. 3(x 2 - 1) = 0 \Rightarrow 3(x - 1)(x + 1) = 0. x = 1 or x = -1. At x = 1, g"(1) = 6.1 = 6 > 0. So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1 3 - 3.1 = 1 - 3 = - 2. And at x = -1, g"( -1) = 6 ( -1) = 6 < 0. So, x = -1 is a point of local minima and value of local minimum is given by g(- 1) = (- 1) 3 - 3(- 1) = 1 + 3 = 2. (iii) h (x) = s i n x + c o s x . 0 < x < \frac{π}{2} A(iii) we have, h(x) = sin x + cos x. \Rightarrow h'(x) = cos x - sin x \Rightarrow h'(x) = -sin x - cos x. At extreme points, h'(x) = 0 cosx - sin x = 0 cosx = sin x \Rightarrow \frac{c o s x}{c o s x} = \frac{s i n x}{c o s x} \Rightarrow 1 = tan x \Rightarrow t a n x = t a n \frac{π}{4} \Rightarrow x = \frac{π}{4} . (iv) f (x) = s i n x - c o s x, 0 < x < 2 π A(iv) we have, f(x) = sin x cos x. \Rightarrow f(x) = cos x + sin x \Rightarrow f(x) = sin x + cos x. At extreme points, f(x) = 0 \Rightarrow cosx + sin x = 0. \Rightarrow \frac{s i n x}{c o s x} = \frac{- c o s x}{c o s x} \Rightarrow t a n x = - 1 = - t a n \frac{π}{4} = t a n (π - \frac{π}{4}) = t a n (2 π - \frac{π}{4}) ∴ x = (π - \frac{π}{4}) IInd quadrate or (2 π - \frac{π}{4}) IIIth quadrate \Rightarrow x = \frac{4 π - π}{4} \frac{8 x - π}{4} \Rightarrow x = \frac{3 π}{4} o r x = \frac{f π}{4} . A t x = \frac{3 π}{4}, f^{''} (\frac{3 π}{4}) = - s i n \frac{3 π}{4} + c o s \frac{3 π}{4} = - s i n (π - \frac{π}{4}) + c o s (π - \frac{π}{4}) = - s i n \frac{π}{4} - c o t \frac{π}{4} (v) f(x) = x 3 - 6 x 2 + 9 x + 15. A(v) we have, f(x) = x 3 - 6 x 2 + 9 x + 15. \Rightarrow f(x) = 3 x 2 - 12 x + 9. \Rightarrow f'(x) = 6x - 12. At extreme point, f'(x) = 0. \Rightarrow 3 x 2 - 12 x + 9 = 0. \Rightarrow x 2 - 4 x + 3 = 0 \Rightarrow x 2 - x - 3 x + 3 = 0 \Rightarrow x(x - 1) -3(x - 1) = 0 \Rightarrow (x - 1)(x - 3) = 0 \Rightarrow x = 1 or x = 3. At, x = 1,f"(1) = 6 \times 1 - 12 = 6 - 12 = - 6 < 0 ∴x = 1 is a point of local maxima and the value of local Maximum is given by f(1) = 1 3 - 6(1) 2 + 9(1) + 15 = 1 - 6 + 9 + 15 = 19. And at x = 3,f"(3) = 6 \times 3 - 12 = 18 - 12 = 6 > 0 ∴x = 3 is a point of local minima and the value of local minimum is given by f (3) = 3 3 - 6(3) 2 + 9(3) + 15. = 27 - 54 + 27 + 15. = 15. (vi) g (x) = \frac{x}{2} + \frac{2}{x}, x > 0 A(vi) We have, g (x) = \frac{x}{2} + \frac{2}{x}, x > 0 \Rightarrow g^{'} (x) = \frac{1}{2} - \frac{2}{x^{2}} \Rightarrow g g^{' z} (x) = \frac{x^{2} - 4}{2 x^{2}} = \frac{(x - 2) (x + 2)}{2 x^{2}} g^{''} (x) = \frac{4}{x^{3}} . At, extreme point, g'(x) = 0 \Rightarrow \frac{(x + 2) (x - 2)}{x^{2}} = 0 \Rightarrow x = -2 or x = 2 Given, that x > 0, hence we have x = 2. At, x = 2, {\overset{?}{g}}^{'} (x) = \frac{4}{2^{3}} = \frac{4}{8} = \frac{1}{2} > 0 ∴x = 2 is a point of local minima and value of local Minimum is given by g(2) = \frac{2}{2} + \frac{2}{2} = 1 + 1 = 2 . (vii) f (x) = \frac{1}{x^{2} + 2} A(vii) Given, f (x) = \frac{1}{x^{2} + 2} \Rightarrow g^{'} (x) = - \frac{1}{{(x^{2} + 2)}^{2}} \frac{d}{d x} (x^{2} + 2) = \frac{- 2 x}{{(x^{2} + 2)}^{2}} . \Rightarrow g^{''} (x) = - [\frac{{(x^{2} + 2)}^{2} \frac{d}{d x} 2 x - 2 x \frac{d}{d x} {(x^{2} + 2)}^{2}}{{(x^{2} + 2)}^{4}}] = \frac{- [2 {(x^{2} + 2)}^{2} - 2 x \cdot 2 (x^{2} + 2) \cdot 2 x]}{{(x^{2} + 2)}^{4}} . = \frac{- [2 {(x^{2} + 2)}^{2} [x^{2} + 2 - 4 x^{2}]]}{{(x^{2} + 2)}^{4}} = \frac{- 2 (x^{2} + 2) (2 - 3 x^{2})}{{(x^{2} + 2)}^{4}} = - \frac{2 (2 - 3 x^{2})}{{(x^{2} + 2)}^{3}} . At extreme points, g'(x) = 0. \Rightarrow \frac{- 2 x}{{(x^{2} + 2)}^{2}} = 0 . \Rightarrow x = 0. At, x = 0, g^{''} (x) = \frac{- 2 (2 - 2 \times 0^{2})}{{(0^{2} + 2)}^{3}} = \frac{- 2 (2 - 0)}{2^{3}} = - \frac{1}{2} < 0 ∴x = 0 is a point of local maxima and value of local maximum is given by g (0) = \frac{1}{0^{2} + 2} = \frac{1}{2} .Q: 78. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x 3, x \in [- 2, 2] Read moreA: (i) We have, f(x) = x 3, x \in [- 2, 2]. f(x) = 3 x 2 . At, f(x) = 0 3 x 2 = 0 x = 0 <--[-2, 2]. We shall absolute the value of f at x = 0 and points of interval [ -2, 2]. So, f(0) = 0 f (- 2) = (- 2) 3 = 8 f (2) = 2 3 = 8. ∴ Absolute maximum value of f(x) = 8 at x = 2 and absolute minimum value of f(x) = -8 at x = -2. (ii) f (x) = sin x + cos x , x \in [0, π] A.(ii) We have, f(x) = sin x + cos x , x \in [0, π] f(x) = cos x - sin x. atf(x) = 0 \Rightarrow cosx - sin x = 0 \Rightarrow sinx = cos x \Rightarrow \frac{s i n^{°} x}{c o r x} = 1 t a n x = 1 \Rightarrow t a n x = t a n \frac{π}{4} \Rightarrow x = \frac{π}{4} \in [0, π] (iii) f(x) = 4x - \frac{1}{2} x^{2}, x \in [- 2, \frac{9}{2}] A.(iii) We have, f(x) = 4x - \frac{1}{2} x^{2}, x \in [- 2, \frac{9}{2}] f(x) = 4 - x atf(x) = 0 \Rightarrow 4- x = 0 \Rightarrow x = 4 \in [-2, \frac{9}{2}] ∴ f (4) = 4 (4) - \frac{1}{2} {(4)}^{2} = 16 - 8 = 8 . f (- 2) = 4 (- 2) - \frac{1}{2} {(- 2)}^{2} = - 8 - 2 = - 10 . f (\frac{9}{2}) = 4 (\frac{9}{2}) - \frac{1}{2} {(\frac{9}{2})}^{2} = 18 - \frac{81}{8} = \frac{94 - 81}{8} = \frac{63}{8} . = 7.87.5 Hence, absolute maximum value of f(x) = 8 at x = 4 and absolute minimum value of f(x) = -10 at x = -2. (iv) f(x) = (x - 1) 2 + 3, x\in [ -3, 1] A.(iv) We have, f(x) = (x - 1) 2 + 3, x\in [ -3, 1] f(x) = 2(x -1) atf(x) = 0 2(x - 1) = 0 x = 1 \in [ -3, 1] ∴f(1) = (1 - 1) 2 + 3 = 3 f (- 3) = (- 3 - 1) 2 + 3 = 16 + 3 = 1 ∴ Absolute maximum value of f(x) = 19 at x = -3. And absolute minimum value of f(x) = 3 at x = 1.Q: 79. Find the ma pro ximum fit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x 2 Read moreA: We have, p (x) = 41 - f 2 x - 18 x 2 . P (x) = - 72 - 36 x P (x) = -36 At extreme point, - 72 - 36 x = 0 \Rightarrow x = \frac{- 72}{36} = - 2 . At x = - 2, p" (x) = - 36 < 0. ∴x = -2 is a point of local maximum and the value of local Maximum is given by P (2) = 41 - 72 (- 2) - 18 (- 2) 2 41 + 144 - 72 = 113 units.Q: 80. Find both the maximum value and the minimum value of 3x 4 - 8x 3 + 12x 2 - 48x + 25 on the interval [0, 3]. Read moreA: We have, f (x) = 3 x 4 - 8 x 3 + 12 x 2 - 48 x + 25, x \in [0, 3]. f (x) = 12 x 3 - 24 x 2 + 24 x - 48. At f (x) = 0. 12 x 3 - 24 x 2 + 24 x - 48 = 0. x 3 - 2 x 2 + 2 x - 4 = 0 x 2 (x - 2) + 2 (x - 2) = 0 (x - 2) + (x 2 + 2) = 0 x = 2 \in [0, 3] or x = \pm\sqrt-2 which is not possible as ∴f (x) = 3 (2) 4 - 8 (2) 3 + 12 (2) 2 - 4 (2) + 25. =48 - 64 + 48 - 96 + 25. = -39. f (0) =3 (0) 4 - 8 (0) 3 + 12 (0) 2 - 48 (0) + 25. = 25. f (3) = 3 (3) 4 - 8 (3) 3 + 12 (3) 2 - 48 (3) + 25. = 243 - 216 + 108 - 144 + 25 = 16. Maximum value of f (x) = 25 at x = 0. and minimum value of f (x) = -39 at x = 2.Q: 81. At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?A: We have, f(x) = sin 2x, x \in [0, 2π], f(x) = 2cos 2x. At f(x) = 0. 2 cos 2x = 0 cos 2x = 0 \Rightarrow 2 x = (2 x + 1) \frac{π}{2}, x = 0, 1, 2, 3 . \Rightarrow x = (2 x + 1) \frac{π}{4} . x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}, \in [0, 2 π] ∴ f (\frac{π}{4}) = s i n \frac{2 π}{4} = s i n \frac{π}{2} = 1 . f (\frac{3 π}{4}) = s i n 2 (\frac{3 π}{4}) = s i n \frac{3 π}{2} f (\frac{7 π}{4}) = s i n (π + \frac{π}{2}) = = s i n 2 \times \frac{7 π}{4} = - s i n \frac{π}{2} = s i n \frac{7 π}{2} = 1. = s i n 3 π + \frac{π}{2} f (\frac{5 π}{4}) = s i n 2 \times (\frac{5 π}{4}) = s i n \frac{5 π}{2} = s i n (2 x + \frac{π}{4}) = - s i n \frac{π}{2} = s i n \frac{π}{4} = 1 = 1. f(0) = sin 2(0) = sin 0 = 0 f(2π) = sin 2(2π) = sin 4π = 0 Hence, the points of maximum x f x are. (\frac{π}{4}, 2)and (\frac{5 π}{4}, 1) .Q: 83. Find the maximum value of 2x 3 - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1] Read moreA: We have, f (x) =2 x 3 - 24 x + 107, x [1,3] f (x) = 6 x 2 - 24. At f (x) = 0 6 x 2 - 24= 0 \Rightarrow x^{2} = \frac{24}{6} = 4 x = \pm2. ->x = 2 \in [1, 3]. So, f (2) = 2 (2) 3 - 24 (2) + 107 = 16 - 48 + 107 = 75. f (1) = 2 (1) 3 - 24 (1) + 107 = 2 - 24 + 107 = 85. f (3) = 2 (3) 3 - 24 (3) + 107 = 54 - 72 + 107 = 89 ∴ Maximum value of f (x) in interval [1, 3] is 89 at x = 3. When x \in [ -3, -1] From f (x) = 0 x = -2 \in [ -3, -1] So, f (- 2) = 2 (- 2) 3 - 24 (- 2) + 107 = - 16 + 48 + 107 = 139. f (- 3) = 2 (- 3) 3 - 24 (- 3) + 107 = - 54 + 72 + 107 = 125. f (- 1) = 2 (- 1) 3 - 24 (- 1) + 107 = - 2 + 24 + 107 = 129. ∴ Maximum value of f (x) in interval [ -3, -1] is 139 at x = -2.Q: 84. It is given that at x = 1, the function x 4 - 62x 2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a. Read moreA: We have, f (x) = x 4 - 62 x 2 + ax + 9, x\in [0, 2]. \Rightarrow f (x) = 4 x 3 - 124 x + a ∴f (x) active its max n value at x = 1 \in [0, 2] ∴f (1) = 0. 4 (1) 3 - 124 (1) + a = 0 \Rightarrow a = 124 - 4 = 120. ∴a = 120Q: 86. Find two numbers whose sum is 24 and whose product is as large as possible.A: Let ‘x’ and ‘y’ be the two number Then, x + y = 24 \Rightarrow y = 24 - x Let ‘P’ be their product then, P = xy = x (24 - x) = 24 x - x 2 P (x) = 24 x - x 2 \Rightarrow \frac{d p}{d x} = 24 - 2 x . \Rightarrow \frac{d^{2} p}{d x^{2}} = - 2 . At \frac{d p}{d x} = 0 24 - 2x = 0 \Rightarrow x = \frac{24}{2} = 12 . So, P (12) \frac{d^{2} p}{d x^{2}} = - 2 < 0 . x = 12 is a point of local maxima Hence, y = 24 - 12 = 12. The uqdtwno (x, y) is (12, 12).Q: 87. Find two positive numbers x and y such that x + y = 60 and xy 3 is maximum.A: We have, x + y = 60.where x, y > 0 x = 60 - y. Let the product P = xy 3 = (60 - y) y 3 = 60 y 3 - y 4 \Rightarrow \frac{d p}{d y} = 180 y^{2} - 4 y^{3} = 4 y 2 (45 - y) At \frac{d p}{d y} = 0 \Rightarrow 4 y 2 (45 - y) = 0 \Rightarrow y = 0 and y = 45 As y > 0, y = 45 When, y > 45, \frac{d p}{d y} = (+ v e) (- v e) = ve < 0 Ad y < 45, \frac{d p}{d y} = (+ v e) (+ v e) = (+ ve) > 0 ∴p is maximum when y = 45 from + ve to- ve or y increases through 45. So, x = 60 - y = 6Ø - 45 = 15. Øx = 15 and y = 45.Q: 88. Find two positive numbers x and y such that their sum is 35 and the product x 2 y 5 is a maximum.A: We have, x + y = 35. y = 35 - x Let the product, P =x 2 y 5 \Rightarrow P = x 2 (35 - x) 5 So, \frac{d p}{d x} = x 2 5 (35 - x) 4 (1) + (35 - x) 52 x = x (35 - x) 4 [ - 5 x + (35 - x) 2] = x (35 - x) 4 [ - 5 x + 70 - 2 x] = x (35 - x) 4 (70 - 7 x) = 7 x (35 - x) 4 (10 - x) At \frac{d p}{d x} = 0 \Rightarrow 7 x (35 - x) 4 (10 - x) = 0 \Rightarrow x = 0, 35, 10 As x is a (+) ve number we have only x = 10, 35 And again (at x = 35) y = 35 = 0 but yis also a (+) ve number we get, x = 10 (only) whenx < 10, ? \frac{d p}{d x} = (+ v e) (+ v e) (+ v e) > 0 and when x > 10, \frac{d p}{d x} = (+ v e) v e (+ v e) (- v e) = (- v e) < 0 ∴ \frac{d p}{d x} changes from (+ ve) to ( -ve) as x increases while passing through 10 Hence, x = 10 is a point of local maxima So, y = 35 - 10 = 25 ∴x = 10 and y = 25Q: 89. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimumA: Let, x and y be the two positive number Then, x + y = 16 y = 16 - x Let p be the sun of the cubes then p = x 3 + y 3 = x 3 + (16 - x) 3 = x 3 + (16) 3 - x 3 - 48 x (16 - x) p = 16 3 + 48 x 2 - 76 8 x So, \frac{d p}{d x} = 96 x - 768 . \Rightarrow \frac{d^{2} p}{d x^{2}} = 96 . At \frac{d p}{d x} = 0 96x - 768 = 0 \Rightarrow x = \frac{768}{96} = 8 . ∴at x = 8, \frac{d^{2} p}{d x^{2}} = 96 \cdot > 0 So, x = 8 is a point of local minima So, y = 16 - 8 = 8 Hence, x = 8, y = 8Q: 90. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible. Read moreA: Side of the tin square piece = 18 cm Let x cm be the thought of the square to be cut from each corner. The volume v of the box after cutting \Rightarrow v = length \times breadth \times height = (18 - 2 x) (18 - 2 x) x x = (18 - 2 x) 2 x = (324 + 4 x 2 - 72 x) x = 4 x 3 - 72 x 2 + 324 x So, \frac{d}{d x} = 12 x^{2} - 144 x + 324 \frac{d^{2} v}{d x^{2}} = 24 x - 144 As \frac{d}{d x} = 0 \to 12 x^{2} - 144 x + 324 = 0 \Rightarrow x 2 - 12 x + 27 = 0 \Rightarrow x 2 - 9 x - 3 x + 27 = 0 \Rightarrow x (x - 9) - 3 (x - 9) = 0 \Rightarrow (x - 9) (x - 3) = 0 \Rightarrow x = 9 and x = 3 At x = 0, length of box = 18 - 2π9 = 18 - 18 = 0 Which is not possible And at x = 3, \frac{d^{2} y}{d x^{2}} = 24 (3) - 144 = -72 < 0 ∴x = 3 is a point of maximum Hence, ‘3’ cm (square) is to be cut from each side of the square So that volume of box is maximumQ: 91. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum? Read moreA: Let ‘x’ cm be the length of side of the square to be cut off from the rectangular surface Then, the volume v of the box is v = (45 - 2 x) (24 - 2 x) x = 1080 x - 138 x 2 + 4 x 3 So, \frac{d v}{d x} = 1080 - 276 x + 12 x^{2} \frac{d^{2}}{d x^{2}} = - 276 + 24 x At, \frac{d v}{d x} = 0 \to 1080 - 276 x + 12 x^{2} = 0 \Rightarrow x 2 - 23 x + 90 = 0 \Rightarrow x 2 - 5 x - 18 x + 90 = 0 \Rightarrow x (x - 5) - 18 (x - 5) = 0 \Rightarrow (x - 5) (x - 18) = 0 \Rightarrow x = 5 and x = 18 At x = 18, breadth = 24 - 2 (18) = 24 - 36 = -12 which is not possible At, x = 5, \frac{d^{2} v}{d x^{2}} = - 276 + 24 (5) = - 276 + 120 = - 150 < 0 Hence, x = 5 is the point of maximum So, ‘5’ cm length of square seeds to be cut from each corner of the secthgleQ: 93. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. Read moreA: Let r and h be the radius and height of the cylinder So, r, h > 0 The total surface area s is given by S = 2πr(h + r) = content . \Rightarrow \frac{S}{2 π} = r (h + r) = content = x (say) \Rightarrow h + r = \frac{u}{r} \Rightarrow h = \frac{k}{r} - r = \frac{k - r^{2}}{r .} Then, the volume v of the cylinder V = π r^{2} h = π r^{2} (\frac{k - r^{2}}{r}) = π r (x - x^{2}) = π (k x - π^{3}) So, \frac{d v}{d r} = π (x - 3 r^{2}) \frac{d^{2}}{d r^{2}} = - 6 π r . For maximum, \frac{d v}{d r} = 0 \Rightarrow π [x - 3 x^{2}) = 0 \Rightarrow r^{2} = \frac{x}{3}Q: 94. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area? Read moreA: The volume v of a cylinder of height h and radius r is V = π r 2 h = 100 \Rightarrow h = \frac{100}{π r^{2}} . Let, s be the surface area then S = 2 r 2 h r(r + h) = 2 x (r^{2} + \frac{100 r}{π r^{2}}) = 2 π (n^{2} + \frac{100}{π n}) . ∴ \frac{d S}{d r} = 2 π [2 x - \frac{100}{1 r^{2}}] \Rightarrow \frac{d^{2} S}{d r^{2}} = 2 π [2 + \frac{200}{π r^{3}}] At, \frac{d s}{d r} = 2 π [2 r - \frac{100}{π r^{2}}] = 0 \Rightarrow 2 r = \frac{100}{π r^{2}} \RightarrowM^{3} = \frac{50}{π} \Rightarrow M = {[\frac{50}{π}]}^{\frac{1}{3}}, > 0 . At, x = {[\frac{50}{π}]}^{\frac{1}{3}}, \frac{d^{2S}}{d r^{2}} = 2 π [2 + \frac{200}{π {[\frac{50}{π}]}^{\frac{1}{3}}} \times 3] = 2 π [2 + 4] = 12 π > 0 ∴ r = {[\frac{50}{π}]}^{\frac{1}{3}} isa point of minimum And h = \frac{100}{π {[\frac{50}{π}]}^{\frac{2}{3}}} = 2 {[\frac{50}{π}]}^{\frac{1}{3}} .Q: 95. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? Read moreA: Let x and y in ‘m’ be the length of side of the square the radius of the circle respectily Then, length of wire = perimeter of square + circumference of circle \Rightarrow 28 = 4x + 2πy \Rightarrow 2x + πy = 14 \Rightarrow y = \frac{14 - 2 x}{x} The combine area A of the square and the circle is A = x 2 + π y 2 = x^{2} + π {[\frac{14 - 2 x}{π}]}^{2} {= x^{2} + \frac{[2 (7 - x)}{x}]}^{2} = x^{2} + \frac{4}{π} {(7 - x)}^{2} . So, \frac{dA}{d x} = 2 x + \frac{4 \cdot 2}{π} (7 - x) + (- 1) = 2 x - \frac{8}{π} (7 - x) \frac{d^{2A}}{d x^{2}} = 2 + \frac{8}{π} At, \frac{dA}{d x} = 0 \Rightarrow 2 x - \frac{8 (7 - x)}{π} = 0 \Rightarrow 2 x = \frac{8}{π} (7 - x) \Rightarrow 2 π x = 56 - 8 x \Rightarrow (2 x + 8) x = 56 \Rightarrow x = \frac{56}{2 π + 8} = \frac{28}{x + 4} . At, x = \frac{28}{π + 4}, \frac{d^{2A}}{d x^{2}} = 2 + \frac{γ}{π} > 0 ∴ x = \frac{28}{π + 4} isa point of minima Hence, length of square = 4 x = 4 (\frac{28}{1 + 4}) = \frac{112}{π + 4} m and length of circle = 2πy = 2 π (\frac{14 - 2 x}{π}) = 2 π [\frac{14}{π} - \frac{2}{1} [\frac{28}{π + 4}]] = 28 - 4 [\frac{28}{π + 4}] = 28 - \frac{112}{1 + 4} = \frac{28 x + 112 - 112}{x + 4} = \frac{28 π}{π + 4}Q: 96. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \frac{8}{27} of the volume of the sphere. Read moreA: Let r and h be the radius and height of the one in scribed in the sphere of radius R. Then, is ΔOBC, rt angle at B (h - r) 2 + r 2 = R 2 \Rightarrow h 2 + R 2 - 2 h R + h 2 = R 2 \Rightarrow r 2 = 2 h R - h 2 Then the volume v of the cone is, V = \frac{1}{3} \times r^{2} h = \frac{1}{3} π (2 h R - h^{2}) h = \frac{1}{3} π (2 {Rh}^{2} - h^{3}) . \Rightarrow \frac{dV}{d h} = \frac{π}{3} (4 Rh - 3 h^{2}) . \frac{d^{2V}}{d h^{2}} = \frac{π}{3} (4R - 6 h) . At \frac{dV}{d h} = 0 - \frac{π}{3} (4R h - 3 h^{2}) = 0 . 4Rh - 3 h 2 = 0. h(4R - 3h) = 0. h = 0 and h = \frac{4 R}{3} As h> 0, h = \frac{4}{3} . At h = \frac{4R}{3}, \frac{d^{2v}}{d h^{2}} = \frac{π}{3} [4 R - 6 \times \frac{4rR}{3}] = \frac{1}{3} [4R - 8 R] = - \frac{4 Rπ}{3} < 0 h = \frac{4R}{3} is a point of maxima. and r^{2} = 2 hR - h^{2} = 2 \times \frac{4R}{3} R- {(\frac{4R}{3})}^{2} = \frac{3 \times 8R^{2}}{9} - \frac{1 6R^{2}}{9} r^{2} = \frac{8R^{2}}{9} Hence, Volume of Cone, = \frac{1}{3} π r^{2} h = \frac{1}{3} π \times \frac{8R^{2}}{9} \times \frac{4R}{3} = \frac{8}{27} \times \frac{4}{3} πR^{3} = \frac{8}{27} \times Volume of sphere.Q: 98. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is t a n^{- 1} \sqrt2. Read moreA: Let r, h, l and Ø be the radius, height, slant height and semi-vertical angle respectively of the cone. i.e., r, h, l>0. Then, Volume V of the cone is = \frac{1}{3} π r^{2} h . = \frac{1}{3} π (l^{2} - h^{2}) h = \frac{1}{3} π (l^{2} h - h^{3}) . So, \frac{dV}{d h} = \frac{1}{3} π (l^{2} - 3 h^{2}) \frac{d^{2V}}{d h^{2}} = - 2 π hQ: 100. The point on the curve x2 = 2y which is nearest to the point (0, 5) is (A) (2\sqrt2,4) (B) (2\sqrt2,0) (C) (0, 0) (D) (2, 2) Read moreA: The equation of the given curve is x 2 = 2y. Let p (x, y) be a point on the curve. The distance of p (x, y) from (0, 5) is say S is given by s^{2} = x^{2} + {(y - 5)}^{2} Let z = s 2 = x 2 + y 2 + 25 - 10 y = 2 y + y 2 -10 y + 25 z = y 2 - 8 y + 25 So, \frac{d z}{d y} = 2 y - 8 \frac{d^{2} z}{d y^{2}} = 2 At \frac{d z}{d y} = 0 \Rightarrow 2 y - 8 = 0 \Rightarrow y = \frac{8}{2} = 4 At y = 4, \frac{d^{2} z}{d y^{2}} = 2 > 0 y = 4, is point of minimum distance. So, x 2 = 2 y -> x 2 = 2 \times 4-> x 2 = 8 x = \pm 2 Hence, the point of the nearest distance are (2\sqrt2, 4) and (- 2\sqrt2, 4) . Option (A) is correct.Q: 101. For all real values of x, the minimum value of \frac{1 - x + x^{2}}{1 + x + x^{2}} is (A) 0 (B) 1 (C) 3 (D) \frac{1}{3}A: We have, f (x) = \frac{1 - x + x^{2}}{1 + x + x^{2}} . f (x) = \frac{(1 + x + x^{2}) \frac{d}{d x} (1 - x + x^{2}) - (1 - x + x^{2}) \frac{d}{d x} (1 + x + 2)}{{(1 + x + x^{2})}^{2}} \Rightarrow f^{'} (x) = \frac{(1 + x + x^{2}) (- 1 + 2 x) - (1 - x + x^{2}) (1 + 2 x)}{{(1 + x + x^{2})}^{2}} = \frac{- 1 + 2 x - x + 2 x^{2} - x^{2} + 2 x^{3} - 1 - 2 x + x + 2 x^{2} - x^{2} - 2 x^{3}}{{(1 + x + x^{2})}^{2}} = \frac{- 2 + 2 x^{2}}{{(1 + x + x^{2})}^{2}} = \frac{- 2 (1 - x^{2})}{{(1 + x + x^{2})}^{2}} = - \frac{2 (1 + x) (1 - x)}{{(1 + x + x^{2})}^{2}} At f(x) = 0. \Rightarrow \frac{- 2 (1 + x) (1 - x)}{{(1 + x + x^{2})}^{2}} = 0 x = 1 and x = -1. At x = 1, f (1) = \frac{1 - 1 + 1}{1 + 1 + 1} = \frac{1}{3} x = 1, f (1) = \frac{1 - 1 + 1}{1 + 1 + 1} = \frac{1}{3} At x = -1, f (- 1) = \frac{1 + 1 + 1}{1 - 1 + 1} = 3 The maximum value of f(x) = \frac{1}{3} x = 1 Hence, option (D) is correct.Q: 102. The maximum value of {[x (x - 1) + 1]}^{\frac{1}{3}}, 0 (A) {\frac{1}{3}}^{\frac{1}{3}} (B) \frac{1}{2} (C) 1 (D) 0 Read moreA: We have, f^{'} (x) = [X(x - 1) + {1]}^{\frac{1}{3}}, 0 \leq x \leq 1 f (x) = \frac{1}{3} {[x^{2} - x + 1]}^{\frac{- 2}{3}} \frac{d}{d x} (x^{2} - x + 1) = \frac{2 x - 1}{3 {(x^{2} - x + 1)}^{\frac{2}{3}}} At f(x) = 0. 2x - 1 = 0 x = \frac{1}{2} \in [0, 1] f (\frac{1}{2}) = {[\frac{1}{2} (\frac{1}{2} - 1) + 1]}^{\frac{1}{3}} = {[\frac{1}{4} - \frac{1}{2} + 1]}^{\frac{1}{3}} . = {[\frac{1 - 2 + 4}{4}]}^{\frac{1}{3}} = {[\frac{3}{4}]}^{\frac{1}{3}} ? 0.90 f (0) = [0(0 - 1) + {1]}^{\frac{1}{3}} = 1 . f (1) = [1 (1 - 1) + 1]^{\frac{1}{3}} = 1 . Option (B) is Hence maximum value of f(x) = at x = 1 and x = 0. Option (c) is correct.Q: 103. Using differentials, find the approximate value of each of the following (a) {(\frac{17}{81})}^{\frac{1}{4}}A: (a) Consider y = x^{\frac{1}{4}} . L e t, x = \frac{16}{81} & ? x = \frac{1}{81} . \begin{array}{l} T h e n, ? y = {(x + ? x)}^{\frac{1}{4}} - x^{\frac{1}{4}} \\ = {(\frac{17}{81})}^{\frac{1}{4}} - {(\frac{16}{81})}^{\frac{1}{4}} \\ = {(\frac{17}{81})}^{\frac{1}{4}} - \frac{2}{3} \\ ∴ {(\frac{17}{81})}^{\frac{1}{4}} = \frac{2}{3} + ? y \end{array} Now, d y is approximately equal to ? y and is given by, \begin{array}{l} d y = (\frac{d y}{d x}) ? x = \frac{1}{4 {(x)}^{\frac{3}{4}}} (? x) (a s, y = x^{\frac{1}{4}}) \\ = \frac{1}{4 {(\frac{16}{81})}^{\frac{3}{4}}} (\frac{1}{81}) = \frac{27}{4 \times 8} \times \frac{1}{81} = \frac{1}{32 \times 3} = \frac{1}{96} = 0.010 \end{array} Hence, the approximate value of {(\frac{17}{81})}^{\frac{1}{4}} is \frac{2}{3} + 0.010 = 0.667 + 0.010 =0.677 (b) {(33)}^{\frac{- 1}{5}} (b) Consider y = x^{\frac{1}{5}} . L e t, x = 32 & ? x = 1 Now, d y is approximately equal to ? y and is given by, \begin{array}{l} d y = (\frac{d y}{d x}) (? x) = \frac{- 1}{5 {(x)}^{\frac{6}{5}}} (? x) (a s, y = x^{\frac{1}{5}}) \\ = - \frac{- 1}{5 {(x)}^{6}} (1) = - \frac{1}{320} = - 0.003 \end{array} Hence, the approximate value of 33^{\frac{1}{5}} is \frac{1}{2} + (- 0.003 = 0.497)Q: 104. Show that the function given by f (x) \frac{l o g x}{x}, has maximum at x = e.A: We have, f(x) \frac{l o g x}{x}, x > 0 . f(x) = \begin{array}{l} \underline{x \frac{d}{d x} l o g x - l o g x \frac{d}{d x} x} \\ x^{2} . \end{array} = \frac{x \times \frac{1}{x} - l o g x}{x^{2}} = \frac{1 - l o g x}{x^{2}} f(x) = \frac{x^{2} \frac{d}{d x} (1 - l o g x) - (1 - l o g x) \frac{d}{d x} x^{2}}{x^{4}} = \frac{x^{2} (- \frac{1}{x}) - (1 - l o g x) (2 x)}{x^{4}} = \frac{- 1 - 2 + 2 l o g x}{x^{3}} = \frac{2 l o g x - 3}{x^{3}} At extreme points, f(x) = 0. \Rightarrow \frac{1 - l o g x}{x^{2}} = 0 . \Rightarrow l o g x = 1 = l o g e . \Rightarrow x = e . At x = e, f"(e) = \frac{2 l o g e - 3}{e^{3}} = \frac{2 - 3}{e^{3}} = - \frac{1}{e^{3}} < 0 x = e is a point of maximum.Q: 105. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ? Read moreA: Let ‘b’ and ‘x’ be the fixed base and equal side of isosceales triangle. Then, \frac{d x}{d t} = - 3 cm/s (Ø decreasing).Q: 106. Find the equation of the normal to curve x 2 = 4y which passes through the point (1, 2)A: Equation of the curve is y^{2} = 4 x . . . . . . . . . . (i) \begin{array}{l} 2 y \frac{d y}{d x} = 4 \\ \Rightarrow \frac{d y}{d x} = \frac{4}{2 y} = \frac{2}{y} \\ ∴ {\frac{d y}{d x}]}_{(1, 2)} = \frac{2}{2} = 1 \end{array} Now, the slope of the normal at point (1, 2) is \frac{- 1}{{\frac{d y}{d x}]}_{(1, 2)}} = \frac{- 1}{1} = 1 ∴ Equation of the normal at (1, 2) is y - 2 = - 1 (x - 1) . \begin{array}{l} \Rightarrow y - 2 = - x + 1 \\ \Rightarrow x + y - 3 = 0 \end{array}Q: 107. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ - aθ cosθ is at a constant distance from the origin. Read moreA: We have x = a c o s θ + a θ s i n θ \begin{array}{l} ∴ \frac{d x}{d θ} = - a s i n θ + a s i n θ + a θ c o s θ = a θ c o s θ \\ y = a s i n θ - a θ c o s θ \\ ∴ \frac{d y}{d θ} = a c o s θ - a c o s θ + a θ s i n θ = a θ s i n θ \\ ∴ \frac{d y}{d x} = \frac{d y}{d θ} . \frac{d θ}{d x} = \frac{a θ s i n θ}{a θ c o s θ} = t a n θ \end{array} ∴ Slope of the normal at any point θ is - \frac{1}{t a n θ} The equation of the normal at a given point (x, y) is given by, \begin{array}{l} y - a s i n θ + a θ c o s θ = \frac{- 1}{t a n θ} (x - a c o s θ - a θ s i n θ) \\ \Rightarrow y s i n θ - a s i n^{2} θ + a θ s i n θ c o s θ = - x c o s θ + a c o s^{2} θ + a θ s i n θ c o s θ \\ \Rightarrow x c o s θ + y s i n θ - a (s i n^{2} θ + c o s^{2} θ) = 0 \\ \Rightarrow x c o s θ + y s i n θ - a = 0 \end{array} Now, the perpendicular distance of the normal from the origin isQ: 108. Find the intervals in which the function f given by f(x)= \frac{4 s i n x - 2 x - x c o s x .}{2 + c o s x} is (i) increasing (ii) decreasing Read moreA: We have, f(x)= \frac{4 s i n x - 2 x - x c o s x .}{2 + c o s x} = \frac{4 s i n x - x (2 + c o s x)}{2 + c o s x} f (x) = \frac{4 s i n x}{2 + c o s x} - x . So, f^{'} (x) = \frac{(2 + c o s x) \frac{d}{d x} (4 s i n x) - 4 s i n x \frac{d}{d x} (2 + c o s x)}{{(2 + c o s x)}^{2}} - \frac{d x}{d x} . = \frac{(2 + c o s x) (4 c o s x) - (4 s i n x) (- s i n x)}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 c o s x + 4 c o s^{2} x + 4 s i n^{2} x}{{(2 + c o s x)}^{2}} - 1 = \frac{8 c o s x + 4 (c o s^{2} x + s i n^{2} x)}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 c o s x + 4}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 \cdot c o s x + 4 - {(2 + c o s x)}^{2}}{{(2 + c o s x)}^{2}} = \frac{8 c o s x + 4 - 4 - 4 c o s x - c o s^{2} x}{{(2 + c o s x)}^{2}} = \frac{4 c o s x - c o s^{2} x}{{(2 + c o s x)}^{2}} = \frac{c o s x (4 - c o s x)}{{(2 + c o s x)}^{2}} . Now, {(2 + c o s x)}^{2} > 0 . And, 4 - c o s x > 0 as cos x lies in [1, 1]. So, (i) for increasing, f(x) \geq 0. cosx \geq 0. x lies in Ist and IVth quadrant. i.e., f(x) is increasing for 0 \leq x \leq \frac{x}{2} and \frac{3 x}{2} \leq x \leq 2 x . (ii) for decreasing, f(x) \leq 0. cosx \leq 0. x lies in IInd and IIIrd quadrant. i.e., f(x) is decreasing for \frac{x}{2} \leq x \leq \frac{3 x}{2} .Q: 109. Find the intervals in which the function f given by f (x) = x^{3} + \frac{1}{x^{3}}, x \neq 0 is (i) Increasing (ii) DecreasingA: We have, f(x) = x^{3} + \frac{1}{x^{3}}, x \neq 0 . \Rightarrow f^{'} (x) = 3 x^{2} - \frac{3 \times 1}{x^{4}} = 3 (x^{2} - \frac{1}{x^{4}}) = 3 (\frac{x^{6} - 1}{x^{4}}) . = \frac{3}{x^{4}} [{(x^{2})}^{3} - 1^{3}] = \frac{3}{x^{4}} (x^{2} - 1) (x^{4} + x^{2} + 1) {(a^{3} - b^{3}) = (a - b) (a^{2} + b^{2} + a b) \Rightarrow f^{'} (x) = \frac{3 (x - 1)}{x^{4}} (x + 1) (x^{4} + x^{2} + 1) . {? x^{2} - b^{2} = (a - b) (a + b) At f^{'} (x) = 0 . \Rightarrow \frac{3 (x - 1) (x + 1) (x^{4} + x^{2} + 1)}{x^{4}} = 0 \Rightarrow x = 1 x = - 1 . 3 (x^{4} + x^{2} + 1) \neq 0 . So we have three disjoint internal i.e., (- \infty, - 1], [- 1, 1] (1, \infty) . When, x \in (- \infty, - 1] . f^{'} (x) = \frac{(+) v e (- v e) (- v e) (+ v e)}{(+ v e)} = (+) v e and 0 at x = - 1 . So, f(x) is increasing. When x \in [- 1, 1] f^{'} (x) = (+ v e) (- v) (+ v e) = (- v e) 0 at x = 1and - 1 So, f(x) is decreasing. When x \in [1, \infty) f(x) = (+ v e) (+ v e) (+ v e) = (+ v e) on 0 at x = 1 So, f(x) is increasing. f(x) is increasing for x \in (\infty,1) and [1, \infty] and decreasing for x \in [1, 1].Q: 110. Find the maximum area of an isosceles triangle inscribed in the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 with its vertex at one end of the major axis. Read moreA: The given equation of the ellipse is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (1) Let the major axis be along x-axis so, vertex is at (\pm a, 0) Let ΔABC be the isosceles triangle inscribed on the ellipse with one vertex C at (a, 0). Then, let A have Co-ordinate (x0, yo) from figure. So, Co-ordinate of B = (x0, y0) As A and B lies on the ellipse, from equation (i),Q: 111. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m 3 . If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank? Read moreA: Let x and y meters be the length and breath of the rectangular base of the tank respectively. Then, volume V of the tank is V = length \times depth \times breath. V = 2xy = 8m 3 (given). \Rightarrow y = \frac{8}{2 x} = \frac{4}{x} . Let ‘t’ be the total cost of building the tank. Then, t = cost of base + cost of sides. = 70xy + 45[4x+4y] {there are four sides. = 70xy + 180x+ 180y. = = 70 \cdot x \cdot \frac{4}{x} + 180 x + 180 \times \frac{4}{x} . t = 280 + 180 x + \frac{720}{x} . So, \frac{d z}{d x} = 0 + 180 - \frac{720}{x^{2}} . And \frac{d^{2} t}{d x^{2}} = \frac{1440}{x^{3}} At \frac{d^{2}}{d x} = 0 \Rightarrow 180 - \frac{720}{x^{2}} = 0 \Rightarrow x^{2} = \frac{720}{180} = 4 x = \pm 2 x = 2, (x = length and it cannot be negative) At x = 2, \frac{d^{2} t}{d x^{2}} = \frac{1440}{2^{3}} = \frac{1440}{8} = 180 > 0 . x = 2 is point of maxima. Hence, minimum cost = 280 + 180 \times 2 + \frac{720}{2} = 280 + 360 + 360 = 1000.Q: 112. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. Read moreA: Let r and s be the radius of the circle and length of side of square. Then, sum of perimeter of circle and square = k 2πr +4s = k s = \frac{k - 2 \times r}{4} The area A be the total areas of the circle and square. Then, A = π r 2 + s 2 = π r^{2} + {(\frac{k - 2 π r}{4})}^{2} = π r^{2} + \frac{x^{2} + 4 π^{2} r^{2} - 4 π r k}{16} = \frac{16 π r^{2} + x^{2} + 4 π^{2} r^{2} - 4 π k \cdot r}{16} = \frac{1}{16} [(16 π + 4 π^{2}) π^{2} - 4 π k r + x^{2}] . So, \frac{d A}{d r} = \frac{1}{16} [(16 π + 4 x^{2}) \cdot 2 n - 4 π k .] . And \frac{d^{2A}}{d r^{2}} = \frac{1}{16} [(6 π + 4 π^{2}) 2] = \frac{16 π + 4 π^{2}}{8} At \frac{d A}{d r} = 0 \frac{1}{16} [(16 x + 4 x^{2}) 2 x - 4 π k] = 0 \Rightarrow (16 x + 4 x^{2}) 2 x - 4 π k = 0 \Rightarrow 4 π [4 + π] 2 r = 4 π k . r = \frac{k}{2 (4 + π) .} At r = \frac{x}{2 (4 + π)}, \frac{d^{2A}}{d x^{2}} = \frac{16 π + 4 π^{2}}{8} > 0 . s = \frac{2 x}{2 (4 + x) .} s = 2r. Hence, proved.Q: 113. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. Read moreA: Let ‘x’ metre be the radius of the semi-circular opening mounded on the length ‘2x’ side of rectangle. Then, let ‘y’ be the breadth of the rectangle. Then, perimeter of the window = 10m \Rightarrow \frac{2 π x}{2} + 2 x + y + y = 10 x + 2x + 2y + = 10. \Rightarrow y = \frac{10 - (π + 2) π}{2 .} Let the area of the window be A. Then, A = \frac{1}{2} (π x^{2}) + 2 x y . = \frac{π x^{2}}{2} + 2 x \cdot [\frac{10 - (x + 2) x}{2}] . = \frac{π x^{2} + 20 x - 2 π x^{2} - 4 x^{2}}{2} = \frac{1}{2} [-π x 2 - 4 x 2 + 20 x]. So, \frac{d}{d x} = \frac{1}{2} [ -2πx - 8x + 20] And \frac{d \overset{2}{}}{d x^{2}} = \frac{1}{2} [ -2π - 8] = -π -4 = -( π+ 4) At \frac{d}{d x} = 0 . \frac{1}{2} [ -2πx - 8x + 20] = 0 2x + 8x = 20 x = \frac{20}{2 π + 8} = \frac{10}{π + 4} . At x = \frac{10}{π + 4}, \frac{d^{2A}}{d x^{2}} = -( π+ 4) < 0 Øx = \frac{10}{x + 4} is a point of minima. And y = \frac{10 - (x + 2) \times (\frac{10}{x + 4})}{2} = \frac{10 (π + 4) - (π + 2) 10}{2 (π + 4)} = \frac{10 π + 40 - 10 π - 20}{2 (x + 4)} = \frac{10}{π + 4 .} Ø Dimensions of the window are length = 2x = \frac{20}{π + 4} m breadth = y \frac{10}{π + 4} m . radius = y = \frac{10}{π + 4} m .Q: 114. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is {(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{3}{2}} . Read moreA: Let P be the point on hypotenuse of a triangle. ABC, t angle at B. Which is at distance a& b from the sides of the triangle. Let < BAC = < MPC = Ø. Then, in\dots ΔANP, \Rightarrow t a n^{3} θ = \frac{a}{b} t a n θ = {(\frac{a}{b})}^{\frac{1}{3} .} At tan Ø = {(\frac{a}{b})}^{\frac{1}{3,}} \frac{d^{2} \overset{?}{q}}{d θ^{2}} = + v e . > 0 . { Øall trigonometric f x n are + ve in I st quadrant}. So, z is least for tanØ = {(\frac{a}{b})}^{\frac{1}{3 .}} As, Sec 2 Ø = 1 + tan 2 Ø = 1 + {(\frac{a}{b})}^{\frac{2}{3 .}} = \frac{b^{\frac{2}{3}} + a^{\frac{2}{3}}}{b^{\frac{2}{3 .}}} secØ = {[\frac{b^{\frac{2}{3}} + a^{\frac{2}{3}}}{b^{\frac{2}{3}}}]}^{\frac{1}{2}} = \frac{{(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{1}{2}}}{b^{\frac{1}{3} .}} And tan 2 Ø = \frac{1}{c o t^{2} θ} cot 2 Ø = {(\frac{a}{b})}^{\frac{2}{3}} And cosec 2 Ø = 1 + cot 2 Ø = 1 + {(\frac{a}{b})}^{\frac{2}{3}} = \frac{a^{\frac{2}{3}} + b^{\frac{2}{3}}}{a^{\frac{2}{3}}} cosecØ = \frac{{(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{1}{2} .}}{a^{\frac{1}{3}}}Q: 115. Find the points at which the function f given by f (x) = (x - 2) 4 (x + 1) 3 has (i) local maxima (ii) local minima (iii) point of inflexion Read moreA: We have, f (x) = (x - 2) 4 (x + 1) 3 . So, f (x) = (x - 2) 4 . 3 (x + 1) 2 + (x + 1) 3 . 4 (x - 2) 3 . = (x - 2) 3 [x + 1) 2 [3 (x - 2) + 4 (x + 1)] = (x - 2) 3 (x + 1) 2 (3 x - 6 + 4 x + 4) = (x - 2) 3 (x + 1) 2 (7 x - 2). At f (x) = 0. (x - 2) 3 (x + 1) 2 . (7 x - 2) = 0. x = 2, x = -1 or x = \frac{2}{7} . As (x + 1) 2 > 0, we shave evaluate for the remaining factor. At x = 2, When x< 2, f (x) = ( -ve) (+ ve) (+ ve) = ( -ve) < 0. When x> 2, f (x) = (+ ve) (+ ve) (+ ve) = (+ ve) > 0. Øf (x) change from ( -ve) to (+ ve) as x increases So, x = 2 is a point of local minima At x = -1. When x< -1, f (x) = ( -ve) (+ ve) ( -ve) =\notin, ve > 0. When x> -1, f (x) = ( -ve) (+ ve) (+ ve) =\notin, ve > 0. So, f (x) does not change through x -1. Hence, x = -1 is a point of infixion At x = \frac{2}{7,} When x< \frac{2}{7}, f (x) = ( -ve) (+ ve) ( -ve) = ( -ve > 0. When x< \frac{2}{7}, f (x) = ( -ve) (+ ve) (+ ve) = ( -ve < 0. f (x) change from (+ ve) to ( -ve) as x increases through \frac{2}{7} . x = \frac{2}{7} is a point of local maximaQ: 116. Find the absolute maximum and minimum values of the function f given by f (x) =cos 2 x + sin x, x \in [0, π] Read moreA: We have, f (x) = cos 2 x + sin x, x\in [0, π ]. So, f (x) = 2 cos x ( -sin x) + cos x = cos x (1 - 2 sin x). At f (x) = 0 cosx (1 - 2 sin x) = 0 cosx = 0 or 1 - 2 sin x = 0 cosx = cos \frac{π}{2} or sin x = \frac{1}{2} = sin \frac{π}{6} = sin x - \frac{π}{6} x= \frac{π}{2}, x = \frac{π}{6} and x = \frac{5 π}{6} [0, π ]. So, f (\frac{π}{2}) = cos 2 \frac{π}{2} + sin \frac{π}{2} = 1. Absolute minimum of f (x) = \frac{5}{4} and absolute minimum of f (x) = 1.Q: 117. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \frac{4 r}{3} . Read moreA: Let x be the radius of the sphere ad x be radius of the right circular cone. Let height of cone = y Then, in ΔOBA, (y - r) 2 + x 2 = r 2 y 2 + r 2 - 2 ry + x 2 = r 2 x 2 =2 ry - y 2 So, the volume V of the cone is V= \frac{1}{3} π \cdot x^{2} \cdot y {= \frac{1}{3} π {(r a d i u s)}^{2} \timeshight} = \frac{1}{3} \times (2 x y - y^{2}) y = \frac{1}{3} x (2 x y^{2} - y^{3}) . So, \frac{d v}{d y} = \frac{π}{3} [4 x y - 3 y^{2}] And \frac{d^{2}}{d y^{2}} = \frac{x}{3} [4 x - 6 y] . At \frac{d V}{d y} = 0 . \Rightarrow \frac{π}{3} [4 x y - 3 y^{2}] = 0 4 x - y - 3 y 2 = 0 \Rightarrow y = \frac{4 x}{3} asy> 0. At y = \frac{4 π}{3}, \frac{d^{2} v}{d y^{2}} = \frac{π}{3} [4 x - 8 x] = - \frac{4 π r}{3} < 0 . Ø V is maximum when y = \frac{4 π}{3}Q: 118. Let f be a function defined on [a, b] such that f '(x) > 0, for all x \in (a, b). Then prove that f is an increasing function on (a, b). Read moreA: We have, f (x) defined on [a, b] And f (x) > 0 \forall x \in [a, b]. Let x 1, x 2 \in [a, b] and x 2 > x 1 In the internal x 1, x 2], f (x) will also be continuous and differentiable. Hence by mean value theorem, there exist c [x 1, x 2] such that f (c) = \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1 .}} f (x) > 0 \forall x \in [a, b]. Then, f (c) > 0. \Rightarrow \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1}} > 0 i.e., f (x 2) - f (x 1) > 0 f (x 2) > f (x 1). Hence, the function f (x) is always increasing on [a, b]Q: 119. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/\sqrt3. Also find the maximum volume. Read moreA: Kindly go through the solutionQ: 121. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h Read moreA: Let x be the depth of the wheat inside the cylindrical tank is with radius = 10 cm Then, Volume V of the cylindrical tank is V = π (10) 2 x = 100πp x m 3 As \frac{d}{d t} = 314 \frac{3}{h} \Rightarrow 100 π \frac{d x}{d t} = 314 \frac{d x}{d t} = \frac{314}{100 π} = \frac{314}{100 π 3.14} = \frac{314}{314} = 1 . \frac{m}{h} i e, rate of increasing of depth option (A) is correctQ: 124. The normal to the curve x 2 = 4y passing (1,2) is (A) x + y = 3 (B) x - y = 3 (C) x + y = 1 (D) x - y = 1 Read moreA: The equation of the given curve is x^{2} = 4 y Differentiating with respect to x, we have: \begin{array}{l} 2 x = 4 . \frac{d y}{d x} \\ \Rightarrow \frac{d y}{d x} = \frac{x}{2} \end{array} The slope of the normal to the given curve at point (h,k) is given by, \frac{- 1}{{\frac{d y}{d x}]}_{(h, k)}} = - \frac{2}{h} ∴ Equation of the normal at point (h,k) is given as: y - k = \frac{- 2}{h} (x - h) Now, it is given that the normal passes through the point (1,2). Therefore, we have: 2 - k = \frac{- 2}{h} (1 - h) o r, k = 2 + \frac{2}{h} (1 - h) . . . . . . . . . . (i) Since (h,k) lies on the curve x^{2} = 4 y,we have h^{2} = 4 k \Rightarrow k = \frac{h^{2}}{4} From equation (i), we have: \begin{array}{l} \frac{h^{2}}{4} = 2 + \frac{2}{h} (1 - h) \\ \Rightarrow \frac{h^{3}}{4} = 2 h + 2 - 2 h = 2 \end{array} \begin{array}{l} \Rightarrow h^{3} = 8 \\ \Rightarrow h = 2 \\ ∴ k = \frac{h^{2}}{4} \Rightarrow k = 1 \end{array} Hence, the equation of the normal is given as: \begin{array}{l} \Rightarrow y - 1 = \frac{- 2}{2} (x - 2) \\ \Rightarrow y - 1 = - (x - 2) \\ \Rightarrow x + y = 3 \end{array} Therefore, option (A) is correct.Q: 125. The points on the curve 9y 2 = x 3, where the normal to the curve makes equal intercepts with the axes are (A) (4, \pm \frac{8}{3}) (B) 4, \frac{- 8}{3} (C) 4, \pm \frac{3}{8} (D) \pm 4, \frac{3}{8} Read moreA: The equation of the given curve is 9 y^{2} = x^{3} Differentiate with respect to x, we have: \begin{array}{l} 9 (2 y) \frac{d y}{d x} = 3 x^{2} \\ \Rightarrow \frac{d y}{d x} = \frac{x^{2}}{6 y} \end{array} The slope of the normal to the given curve at point (x_{1}, y_{1}) is \frac{- 1}{{\frac{d y}{d x}]}_{(x_{1}, y_{1})}} = - \frac{6 y_{1}}{{x_{1}}^{2}} ∴ The equation of the normal to the curve at (x_{1}, y_{1}) is \begin{array}{l} y - y_{1} = - \frac{6 y_{1}}{{x_{1}}^{2}} (x - x_{1}) \\ \Rightarrow {x_{1}}^{2} y - {x_{1}}^{2} y_{1} = - 6 x y_{1} + 6 x_{1} y_{1} \\ \Rightarrow 6 x y_{1} + {x_{1}}^{2} y = 6 x_{1} y_{1} + {x_{1}}^{2} y_{1} \\ \Rightarrow \frac{6 x y_{1}}{6 x_{1} y_{1} + {x_{1}}^{2} y_{1}} + \frac{{x_{1}}^{2} y}{6 x_{1} y_{1} + {x_{1}}^{2} y_{1}} = 1 \\ \Rightarrow \frac{x}{\frac{x_{1} (6 + x_{1})}{6}} + \frac{y}{\frac{y_{1} (6 + x_{1})}{x_{1}}} \end{array} It is given that the normal makes intercepts with the axes. Therefore, we have: \begin{array}{l} ∴ \frac{x_{1} (6 + x_{1})}{6} = \frac{y_{1} (6 + x_{1})}{x_{1}} \\ \Rightarrow \frac{x_{1}}{6} = \frac{y_{1}}{x_{1}} \\ \Rightarrow {x_{1}}^{2} = 6 y_{1} . . . . . . . . . . (i) \end{array} Also, the point (x_{1}, y_{1}) lies on the curve, so we have 9 {y_{1}}^{2} = {x_{1}}^{3} . . . . . . . . . . (i i) From (i) and (ii), we have: 9 (\frac{{x_{1}}^{2}}{6}) = {x_{1}}^{3} \Rightarrow \frac{{x_{1}}^{4}}{4} = {x_{1}}^{3} \Rightarrow x_{1} = 4 From (ii), we have: \begin{array}{l} 9 {y_{1}}^{2} = {(4)}^{3} = 64 \\ \Rightarrow {y_{1}}^{2} = \frac{64}{9} \\ \Rightarrow y_{1} = \pm \frac{8}{3} \end{array} Hence, the required points are (4, \pm \frac{8}{3}) . Therefore, option (A) is correct.Q: 36. Prove that the function given by f (x) = x 3 - 3x 2 + 3x - 100 is increasing in R.A: We have, f (x) = x 3 - 3 x 23 x - 100. So, f (x) = 3 x 2 - 6 x + 3 = 3 (x 2 - 2 x + 1) = 3 (x - 1) 2 For x \in R . (x - 10) 2 \geq 0, 0 for x = 1. \Rightarrow 3 (x - 1) 2 \geq 0 \Rightarrow f (x) \geq 0 ∴f (x) is increasing on ?Q: 77. Prove that the following functions do not have maxima or minima: (i) f(x) = e xA: (i) We have, f (x) = e x \Rightarrow f (x) = e x . \Rightarrow f^{'} (x) = e^{x .} At, extreme points, f (x) = 0 \Rightarrow e x = 0 which has no real ‘a’ value ∴f (x) has with maximum or minima (ii) g (x) = log x A (ii) We have, g (x) = log x, \Rightarrow g (x) = \frac{1}{x} At extreme points, g (x) = 0 \to \frac{1}{x} = 0 . \Rightarrow 1 = 0 which is not true. ∴g (x) was value minima or maxima (iii) h (x) = x 3 + x 2 + x + 1. A (iii) We have, h (x) = x 3 + x 2 + x + 1. h (x) = 3 x 2 + 2x + 1 At extreme points, h (x) = 0 \Rightarrow 3 x 2 + 2x + 1 = 0NCERT Solutions for Class 12 Maths Ch 6 Application of Derivatives- FAQs Check few of the most asked questions related to this chapter by students below;Class 12 Application of Derivatives 6.5 Solutions- (Old NCERT) Class 12 Math Application of Derivatives Exercise 6.5 mainly focuses on problems such as finding minimum and maximum values of various types of functions, finding minima and maxima on a given point of functions, real real-world problems ( finding minimum and maximum profit), and finding values of volume, area, and more in a given interval for function. The Class 12 Exercise 6.5 Solutions will guide students about approaches, critical thinking skills, first and second derivative tests, and their applications in optimization. Application of Derivatives Exercise 6.5 consists of 29 Questions including 15 Long, 11 Short, and 3 MCQs. Students can check the complete solutions of the exercise 6.5 below; Application of Derivative Exercise 6.5 Solutions Q74.Find the maximum and minimum values, if any, of the following functions given by. (i) f (x) = {(2 x - 1)}^{2} + 3 A(i) We have, f(x) = (2 x - 1) 2 + 3. For all x \in ℝ, (2 x - 1)^{2} \geq 0 (2 x - 1) 2 + 3 \geq 3. f(x) \geq 3. ∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = \frac{1}{2} Again as x \to \infty, f (x) \to \infty as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values. (ii) f (x) = 9 x^{2} + 12 x + 2 A(ii) We have, f(x) = 9 2 + 12x + 2. f (x) = 9 [x^{2} + \frac{12 x}{9} + \frac{2}{9}] (Taking 9 common from each team). f (x) = 9 [x^{2} + \frac{4 x}{3} + \frac{2}{9}] f (x) = 9 [x^{2} + 2 \times \frac{2 x}{3} + {(\frac{2}{3})}^{2} - {(\frac{2}{3})}^{2} + \frac{2}{9}] f (x) = 9 [{(x + \frac{2}{3})}^{2} - \frac{4}{9} + \frac{2}{9}] f (x) = 9 [\cdot {(x + \frac{2}{3})}^{2} - \frac{2}{9}] = 9 {(x + \frac{2}{3})}^{2} - 2 For all x \in ℝ, {(x + \frac{2}{3})}^{2} \geq 0 \Rightarrow {(x + \frac{2}{3})}^{2} - 2 \geq \cdot 0 - 2 \Rightarrow f(x)\geq - 2. ∴The minimum value of f(x) = -2 when x + \frac{2}{3} = 0 And as x \to \infty, f (x) \to \infty so f(x) has \Rightarrow x = - \frac{2}{3} . no maximum values. (iii) f(x) = (x - 1) 2 + 10 A(iii) we have, f(x) = - (x - 1) 2 + 10 For all x \in ℝ, (x - 1)^{2} \geq 0 . \Rightarrow (x - 1) 2 \leq 0 \Rightarrow -(x - 1) 2 + 10 \leq 10. \Rightarrow f(x) \leq 10. ∴maximum value of f(x) = 10 when x - 1 = 0 \Rightarrow x = 1. And minimum value of f(x) does not exist. (iv) g(x) = x 3 + 1 A(iv) we have, g(x) = x 3 + 1 For the given fx n, x \in ℝ, g (x) \to \infty a s x \to \infty and g (x) - \infty a s x \to - \infty Q Maximum and minimum value does not exist. Q75. Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = |x + 2| - 1 A(i) we have, f(x) = |x + 2| - 1 We know that, for all x \in ℝ, | x + 2 | \geq 0 \Rightarrow | x + 2 | - 1 \geq - 1 . \Rightarrow f(x) \geq - 1. ∴ Minimum value of f(x) = -1 when x + 2 = 0 \Rightarrow x = - 2. And maximum value of f(x) does not exist. (ii) g (x) = - | x + 1 | + 3 A(ii) We have, g (x) = - | x + 1 | + 3 For all x \in, | x + 1 | \geq 0 . \Rightarrow - | x + 1 | \leq 0 \Rightarrow - | x + 1 | + 3 \leq 0 + 3 \Rightarrow g(x) \leq 3. ∴ Maximum value of g(x) = 3 when | x + 1 | = 0 \Rightarrow x = - 1 . And minimum value does not exist. (iii) h(x) = sin (2x) + 5. A(iii) we have, h(x) = sin (2x) + 5. For all x \in ℝ, - 1 \leq s i n 2 x \leq 1 . {range of sine function is [-1, 1]} \Rightarrow -1 + 5 \leq sin 2x + 5 \leq 1 + 5. \Rightarrow 4 \leq h(x) \leq 6. ∴ Maximum value of h(x) = 6. Minimum value of h(x) = 4. (iv) f (x) = | s i n 4 x + 3 | . A(iv) we have, f (x) = | s i n 4 x + 3 | . As for all x \in ℝ, - 1 \leq s i n 4 x \leq 1 \Rightarrow -1 + 3 \leq sin 4x + 3 \leq 1 + 3 \Rightarrow | 2 | \leq | s i n 4 x + 3 | \leq | 4 | . \Rightarrow 2 \leq f(x) \leq 4. ∴ Maximum value of f(x) = 4. Minimum value of f(x) = 2. (v) h(x) = x + 1.,x\in ( -1, 1). A(v) we have, h(x) = x + 1.,x\in ( -1, 1). Given, -1 - 1 + 1 < x + 1 < 1 + 1 0 < h (x) < 2. ∴ Maximum value and minimum value of h(x) does not exist. Q76. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) f(x) = x 2 . A(i) we have, f(x) = x 2 . f(x) = 2x. andf(x) = 2. AR extreme point, f(x) = 0 2x = 0 x = 0. When x = 0, f(0) = 2 > 0. ∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0 2 = 0. (ii) g(x) = x 3 3x A(ii) we have, g(x) = x 3 - 3 x \Rightarrow g'(x) = 3 x 2 - 3 \Rightarrow g''(x) = 6x. At extreme point, g'(x) = 0 3 x 2 - 3 = 0. 3(x 2 - 1) = 0 \Rightarrow 3(x - 1)(x + 1) = 0. x = 1 or x = -1. At x = 1, g"(1) = 6.1 = 6 > 0. So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1 3 - 3.1 = 1 - 3 = - 2. And at x = -1, g"( -1) = 6 ( -1) = 6 < 0. So, x = -1 is a point of local minima and value of local minimum is given by g(- 1) = (- 1) 3 - 3(- 1) = 1 + 3 = 2. (iii) h (x) = s i n x + c o s x . 0 < x < \frac{π}{2} A(iii) we have, h(x) = sin x + cos x. \Rightarrow h'(x) = cos x - sin x \Rightarrow h'(x) = -sin x - cos x. At extreme points, h'(x) = 0 cosx - sin x = 0 cosx = sin x \Rightarrow \frac{c o s x}{c o s x} = \frac{s i n x}{c o s x} \Rightarrow 1 = tan x \Rightarrow t a n x = t a n \frac{π}{4} \Rightarrow x = \frac{π}{4} . (iv) f (x) = s i n x - c o s x, 0 < x < 2 π A(iv) we have, f(x) = sin x cos x. \Rightarrow f(x) = cos x + sin x \Rightarrow f(x) = sin x + cos x. At extreme points, f(x) = 0 \Rightarrow cosx + sin x = 0. \Rightarrow \frac{s i n x}{c o s x} = \frac{- c o s x}{c o s x} \Rightarrow t a n x = - 1 = - t a n \frac{π}{4} = t a n (π - \frac{π}{4}) = t a n (2 π - \frac{π}{4}) ∴ x = (π - \frac{π}{4}) IInd quadrate or (2 π - \frac{π}{4}) IIIth quadrate \Rightarrow x = \frac{4 π - π}{4} \frac{8 x - π}{4} \Rightarrow x = \frac{3 π}{4} o r x = \frac{f π}{4} . A t x = \frac{3 π}{4}, f^{''} (\frac{3 π}{4}) = - s i n \frac{3 π}{4} + c o s \frac{3 π}{4} = - s i n (π - \frac{π}{4}) + c o s (π - \frac{π}{4}) = - s i n \frac{π}{4} - c o t \frac{π}{4} (v) f(x) = x 3 - 6 x 2 + 9 x + 15. A(v) we have, f(x) = x 3 - 6 x 2 + 9 x + 15. \Rightarrow f(x) = 3 x 2 - 12 x + 9. \Rightarrow f'(x) = 6x - 12. At extreme point, f'(x) = 0. \Rightarrow 3 x 2 - 12 x + 9 = 0. \Rightarrow x 2 - 4 x + 3 = 0 \Rightarrow x 2 - x - 3 x + 3 = 0 \Rightarrow x(x - 1) -3(x - 1) = 0 \Rightarrow (x - 1)(x - 3) = 0 \Rightarrow x = 1 or x = 3. At, x = 1,f"(1) = 6 \times 1 - 12 = 6 - 12 = - 6 < 0 ∴x = 1 is a point of local maxima and the value of local Maximum is given by f(1) = 1 3 - 6(1) 2 + 9(1) + 15 = 1 - 6 + 9 + 15 = 19. And at x = 3,f"(3) = 6 \times 3 - 12 = 18 - 12 = 6 > 0 ∴x = 3 is a point of local minima and the value of local minimum is given by f (3) = 3 3 - 6(3) 2 + 9(3) + 15. = 27 - 54 + 27 + 15. = 15. (vi) g (x) = \frac{x}{2} + \frac{2}{x}, x > 0 A(vi) We have, g (x) = \frac{x}{2} + \frac{2}{x}, x > 0 \Rightarrow g^{'} (x) = \frac{1}{2} - \frac{2}{x^{2}} \Rightarrow g g^{' z} (x) = \frac{x^{2} - 4}{2 x^{2}} = \frac{(x - 2) (x + 2)}{2 x^{2}} g^{''} (x) = \frac{4}{x^{3}} . At, extreme point, g'(x) = 0 \Rightarrow \frac{(x + 2) (x - 2)}{x^{2}} = 0 \Rightarrow x = -2 or x = 2 Given, that x > 0, hence we have x = 2. At, x = 2, {\dot{g}}^{'} (x) = \frac{4}{2^{3}} = \frac{4}{8} = \frac{1}{2} > 0 ∴x = 2 is a point of local minima and value of local Minimum is given by g(2) = \frac{2}{2} + \frac{2}{2} = 1 + 1 = 2 . (vii) f (x) = \frac{1}{x^{2} + 2} A(vii) Given, f (x) = \frac{1}{x^{2} + 2} \Rightarrow g^{'} (x) = - \frac{1}{{(x^{2} + 2)}^{2}} \frac{d}{d x} (x^{2} + 2) = \frac{- 2 x}{{(x^{2} + 2)}^{2}} . \Rightarrow g^{''} (x) = - [\frac{{(x^{2} + 2)}^{2} \frac{d}{d x} 2 x - 2 x \frac{d}{d x} {(x^{2} + 2)}^{2}}{{(x^{2} + 2)}^{4}}] = \frac{- [2 {(x^{2} + 2)}^{2} - 2 x \cdot 2 (x^{2} + 2) \cdot 2 x]}{{(x^{2} + 2)}^{4}} . = \frac{- [2 {(x^{2} + 2)}^{2} [x^{2} + 2 - 4 x^{2}]]}{{(x^{2} + 2)}^{4}} = \frac{- 2 (x^{2} + 2) (2 - 3 x^{2})}{{(x^{2} + 2)}^{4}} = - \frac{2 (2 - 3 x^{2})}{{(x^{2} + 2)}^{3}} . At extreme points, g'(x) = 0. \Rightarrow \frac{- 2 x}{{(x^{2} + 2)}^{2}} = 0 . \Rightarrow x = 0. At, x = 0, g^{''} (x) = \frac{- 2 (2 - 2 \times 0^{2})}{{(0^{2} + 2)}^{3}} = \frac{- 2 (2 - 0)}{2^{3}} = - \frac{1}{2} < 0 ∴x = 0 is a point of local maxima and value of local maximum is given by g (0) = \frac{1}{0^{2} + 2} = \frac{1}{2} . Q77. Prove that the following functions do not have maxima or minima: (i) f(x) = e x A(i) We have, f(x) = e x \Rightarrow f(x) = e x . \Rightarrow f^{'} (x) = e^{x .} At, extreme points, f(x) = 0 \Rightarrow e x = 0 which has no real ‘a’ value ∴f(x) has with maximum or minima (ii) g(x) = log x A(ii) We have, g(x) = log x, \Rightarrow g(x) = \frac{1}{x} At extreme points, g(x) = 0 \to \frac{1}{x} = 0 . \Rightarrow 1 = 0 which is not true. ∴g(x) was value minima or maxima (iii) h(x) = x 3 + x 2 + x + 1. A(iii) We have, h(x) = x 3 + x 2 + x + 1. h(x) = 3 x 2 + 2x + 1 At extreme points, h(x) = 0 \Rightarrow 3 x 2 + 2x + 1 = 0 Q78. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x 3, x \in [- 2, 2] A.(i) We have, f(x) = x 3, x \in [- 2, 2]. f(x) = 3 x 2 . At, f(x) = 0 3 x 2 = 0 x = 0 <--[-2, 2]. We shall absolute the value of f at x = 0 and points of interval [ -2, 2]. So, f(0) = 0 f (- 2) = (- 2) 3 = 8 f (2) = 2 3 = 8. ∴ Absolute maximum value of f(x) = 8 at x = 2 and absolute minimum value of f(x) = -8 at x = -2. (ii) f (x) = sin x + cos x , x \in [0, π] A.(ii) We have, f(x) = sin x + cos x , x \in [0, π] f(x) = cos x - sin x. atf(x) = 0 \Rightarrow cosx - sin x = 0 \Rightarrow sinx = cos x \Rightarrow \frac{s i n^{°} x}{c o r x} = 1 t a n x = 1 \Rightarrow t a n x = t a n \frac{π}{4} \Rightarrow x = \frac{π}{4} \in [0, π] (iii) f(x) = 4x - \frac{1}{2} x^{2}, x \in [- 2, \frac{9}{2}] A.(iii) We have, f(x) = 4x - \frac{1}{2} x^{2}, x \in [- 2, \frac{9}{2}] f(x) = 4 - x atf(x) = 0 \Rightarrow 4- x = 0 \Rightarrow x = 4 \in [-2, \frac{9}{2}] ∴ f (4) = 4 (4) - \frac{1}{2} {(4)}^{2} = 16 - 8 = 8 . f (- 2) = 4 (- 2) - \frac{1}{2} {(- 2)}^{2} = - 8 - 2 = - 10 . f (\frac{9}{2}) = 4 (\frac{9}{2}) - \frac{1}{2} {(\frac{9}{2})}^{2} = 18 - \frac{81}{8} = \frac{94 - 81}{8} = \frac{63}{8} . = 7.87.5 Hence, absolute maximum value of f(x) = 8 at x = 4 and absolute minimum value of f(x) = -10 at x = -2. (iv) f(x) = (x - 1) 2 + 3, x\in [ -3, 1] A.(iv) We have, f(x) = (x - 1) 2 + 3, x\in [ -3, 1] f(x) = 2(x -1) atf(x) = 0 2(x - 1) = 0 x = 1 \in [ -3, 1] ∴f(1) = (1 - 1) 2 + 3 = 3 f (- 3) = (- 3 - 1) 2 + 3 = 16 + 3 = 1 ∴ Absolute maximum value of f(x) = 19 at x = -3. And absolute minimum value of f(x) = 3 at x = 1. Q79. Find the ma pro ximum fit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x 2 A.6. We have, p(x) = 41 - f 2 x - 18 x 2 . P(x) = - 72 - 36 x P(x) = -36 At extreme point, - 72 - 36 x = 0 \Rightarrow x = \frac{- 72}{36} = - 2 . At x = - 2, p" (x) = - 36 < 0. ∴x = -2 is a point of local maximum and the value of local Maximum is given by P(2) = 41 - 72(- 2) - 18(- 2) 2 41 + 144 - 72 = 113 units. Q80. Find both the maximum value and the minimum value of 3x 4 - 8x 3 + 12x 2 - 48x + 25 on the interval [0, 3]. A7. We have, f(x) = 3 x 4 - 8 x 3 + 12 x 2 - 48 x + 25, x \in [0, 3]. f(x) = 12 x 3 - 24 x 2 + 24 x - 48. At f(x) = 0. 12 x 3 - 24 x 2 + 24 x - 48 = 0. x 3 - 2 x 2 + 2 x - 4 = 0 x 2 (x - 2) + 2(x - 2) = 0 (x - 2) + (x 2 + 2) = 0 x = 2 \in [0, 3] or x = \pm\sqrt-2 which is not possible as ∴f(x) = 3(2) 4 - 8(2) 3 + 12(2) 2 - 4(2) + 25. =48 - 64 + 48 - 96 + 25. = -39. f(0) =3(0) 4 - 8(0) 3 + 12(0) 2 - 48(0) + 25. = 25. f(3) = 3(3) 4 - 8(3) 3 + 12(3) 2 - 48(3) + 25. = 243 - 216 + 108 - 144 + 25 = 16. Maximum value of f(x) = 25 at x = 0. and minimum value of f(x) = -39 at x = 2. Q81. At what points in the interval [0, 2π], does the function sin 2x attain its maximum value? A8. We have, f(x) = sin 2x, x \in [0, 2π], f(x) = 2cos 2x. At f(x) = 0. 2 cos 2x = 0 cos 2x = 0 \Rightarrow 2 x = (2 x + 1) \frac{π}{2}, x = 0, 1, 2, 3 . \Rightarrow x = (2 x + 1) \frac{π}{4} . x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}, \in [0, 2 π] ∴ f (\frac{π}{4}) = s i n \frac{2 π}{4} = s i n \frac{π}{2} = 1 . f (\frac{3 π}{4}) = s i n 2 (\frac{3 π}{4}) = s i n \frac{3 π}{2} f (\frac{7 π}{4}) = s i n (π + \frac{π}{2}) = = s i n 2 \times \frac{7 π}{4} = - s i n \frac{π}{2} = s i n \frac{7 π}{2} = 1. = s i n 3 π + \frac{π}{2} f (\frac{5 π}{4}) = s i n 2 \times (\frac{5 π}{4}) = s i n \frac{5 π}{2} = s i n (2 x + \frac{π}{4}) = - s i n \frac{π}{2} = s i n \frac{π}{4} = 1 = 1. f(0) = sin 2(0) = sin 0 = 0 f(2π) = sin 2(2π) = sin 4π = 0 Hence, the points of maximum x f x are. (\frac{π}{4}, 2)and (\frac{5 π}{4}, 1) . Q82. What is the maximum value of the function sin x + cos x? A9. We have, f(x) = sin x + cos x. f(x) = cos x - sin x. At f(x) = 0 cosx - sin x = 0 sinx = cos x \Rightarrow \frac{s i n x}{c o s x} = 1 . \Rightarrow t a n x = 1 = t a n \frac{π}{4} \Rightarrow x = \frac{π}{4} o r x = n π + \frac{π}{4} . At x = \frac{π}{4} + n π, f (n π + \frac{π}{4}) = s i n (n π + \frac{π}{4}) + c o s (n x + \frac{π}{4}) . = {(- 1)}^{x} s i n \frac{π}{4} + {(- 1)}^{n} s i n \frac{π}{4} . Q83. Find the maximum value of 2x 3 - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1] A.10. We have, f(x) =2 x 3 - 24 x + 107, x [1,3] f(x) = 6 x 2 - 24. At f(x) = 0 6 x 2 - 24= 0 \Rightarrow x^{2} = \frac{24}{6} = 4 x = \pm2. -->x = 2 \in[1, 3]. So, f(2) = 2(2) 3 - 24(2) + 107 = 16 - 48 + 107 = 75. f(1) = 2(1) 3 - 24(1) + 107 = 2 - 24 + 107 = 85. f(3) = 2(3) 3 - 24(3) + 107 = 54 - 72 + 107 = 89 ∴ Maximum value of f(x) in interval [1, 3] is 89 at x = 3. When x \in [ -3, -1] From f(x) = 0 x = -2 \in [ -3, -1] So, f (- 2) = 2(- 2) 3 - 24(- 2) + 107 = - 16 + 48 + 107 = 139. f (- 3) = 2(- 3) 3 - 24(- 3) + 107 = - 54 + 72 + 107 = 125. f (- 1) = 2(- 1) 3 - 24(- 1) + 107 = - 2 + 24 + 107 = 129. ∴ Maximum value of f(x) in interval [ -3, -1] is 139 at x = -2. Q84. It is given that at x = 1, the function x 4 - 62x 2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a. A.11. We have, f(x) = x 4 - 62 x 2 + ax + 9, x\in [0, 2]. \Rightarrow f(x) = 4 x 3 - 124 x + a ∴f(x) active its max n value at x = 1 \in [0, 2] ∴f(1) = 0. 4(1) 3 - 124(1) + a = 0 \Rightarrow a = 124 - 4 = 120. ∴a = 120 Q85. Find the maximum and minimum values of x + sin 2x on [0, 2π]. A.12. We have, f(x) = x + sin 2x ,x \in [0, 2π]. f(x) = 1 + 2cos 2x At f(x) = 0 1 + 2 cos2x = 0 \Rightarrow c o s 2 x = - \frac{1}{2} = - c o s \frac{π}{3} = c o s π - \frac{π}{3} = c o s \frac{2 π}{3} ∴ 2 x = 2 n π \pm \frac{2 π}{3}, n = 1, 2, 3 . \Rightarrow x = n π \pm \frac{2 π}{3} n = 0, x = \pm \frac{π}{3} \to x = \frac{π}{3} \in [0, 2 π] . n = 1, x = π \pm \frac{π}{3} \to x = π + \frac{π}{3} a n d π - \frac{π}{3} = \frac{4 π}{3} a d \frac{2 π}{3} \in [0, 2 π] n = 2, x = 2 π \pm \frac{π}{3} \to x = 2 π + \frac{π}{3} a d 2 π - \frac{π}{3} . \Rightarrow x = \frac{51}{3} \in [0, 2 π] . Hence, x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3} a n d \frac{5 π}{3} Missing At x = \frac{π}{3}, f (\frac{π}{3}) = \frac{π}{3} + s i n \frac{2 π}{3} = 1.05 + s i n (π - \frac{π}{3}) = 1.05 + s i n \frac{π}{3} = 1.05 +\sqrt3/2 = 1.05 + 0.87 = 1.92 At x = \frac{2 π}{3}, f (\frac{2 π}{3}) = \frac{2 π}{3} + s i n 2 \times \frac{2 π}{3} = 2.10 + s i n (π + \frac{π}{3}) = 2.10 - s i n \frac{π}{3} = 2, 10 - 0.87 = 1.23 At x = \frac{4 π}{3}, f (\frac{4 π}{3}) = \frac{4 π}{3} + s i n 2 \times \frac{4 π}{3} = 4 \cdot 2 + s i n (3 x - \frac{π}{3}) = 4.2 + s i n \frac{π}{3} = 4.2 + 0.87 . =5.07. At x = \frac{5 π}{3}, f (\frac{5 π}{3}) = \frac{5 π}{3} + s i n 2 \times \frac{5 π}{3} = 5.25 + s i n (3 x + \frac{π}{3}) = 5 \cdot 25 - s i n \frac{1}{3} = 5.25 - 0.87 = 4.38 At and points, f(0) = 0 + sin2 \times 0 = 0 f(2π) = 2π + sin 2 \times 2π = 6.2 + 0 = 6.28 ∴Maximum value of f(x) = 6.28 at x = 2π and minimum value of f(x) = 0 at x= 0 Q86. Find two numbers whose sum is 24 and whose product is as large as possible. A.13. Let ‘x’ and ‘y’ be the two number Then, x + y = 24 \Rightarrow y = 24 - x Let ‘P’ be their product then, P = xy = x (24 - x) = 24 x - x 2 P(x) = 24 x - x 2 \Rightarrow \frac{d p}{d x} = 24 - 2 x . \Rightarrow \frac{d^{2} p}{d x^{2}} = - 2 . At \frac{d p}{d x} = 0 24 - 2x = 0 \Rightarrow x = \frac{24}{2} = 12 . So, P(12) \frac{d^{2} p}{d x^{2}} = - 2 < 0 . x = 12 is a point of local maxima Hence, y = 24 - 12 = 12. The uqdtwno (x, y) is (12, 12). Q87. Find two positive numbers x and y such that x + y = 60 and xy 3 is maximum. A.14. We have, x + y = 60.where x, y > 0 x = 60 - y. Let the product P = xy 3 = (60 - y) y 3 = 60 y 3 - y 4 \Rightarrow \frac{d p}{d y} = 180 y^{2} - 4 y^{3} = 4 y 2 (45 - y) At \frac{d p}{d y} = 0 \Rightarrow 4 y 2 (45 - y) = 0 \Rightarrow y = 0 and y = 45 As y > 0, y = 45 When, y > 45, \frac{d p}{d y} = (+ v e) (- v e) = ve < 0 Ad y < 45, \frac{d p}{d y} = (+ v e) (+ v e) = (+ ve) > 0 ∴p is maximum when y = 45 from + ve to- ve or y increases through 45. So, x = 60 - y = 6Ø - 45 = 15. Øx = 15 and y = 45. Q88. Find two positive numbers x and y such that their sum is 35 and the product x 2 y 5 is a maximum. A.15. We have, x + y = 35. y = 35 - x Let the product, P =x 2 y 5 \Rightarrow P = x 2 (35 - x) 5 So, \frac{d p}{d x} = x 2 5(35 - x) 4 (1) + (35 - x) 52 x = x (35 - x) 4 [ - 5 x + (35 - x) 2] = x (35 - x) 4 [ - 5 x + 70 - 2 x] = x (35 - x) 4 (70 - 7 x) = 7 x (35 - x) 4 (10 - x) At \frac{d p}{d x} = 0 \Rightarrow 7 x (35 - x) 4 (10 - x) = 0 \Rightarrow x = 0, 35, 10 As x is a (+) ve number we have only x = 10, 35 And again (at x = 35) y = 35 = 0 but yis also a (+) ve number we get, x = 10 (only) whenx < 10, ∵ \frac{d p}{d x} = (+ v e) (+ v e) (+ v e) > 0 and when x > 10, \frac{d p}{d x} = (+ v e) v e (+ v e) (- v e) = (- v e) < 0 ∴ \frac{d p}{d x} changes from (+ ve) to ( -ve) as x increases while passing through 10 Hence, x = 10 is a point of local maxima So, y = 35 - 10 = 25 ∴x = 10 and y = 25 Q89. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum A.16. Let, x and y be the two positive number Then, x + y = 16 y = 16 - x Let p be the sun of the cubes then p = x 3 + y 3 = x 3 + (16 - x) 3 = x 3 + (16) 3 - x 3 - 48 x (16 - x) p = 16 3 + 48 x 2 - 76 8 x So, \frac{d p}{d x} = 96 x - 768 . \Rightarrow \frac{d^{2} p}{d x^{2}} = 96 . At \frac{d p}{d x} = 0 96x - 768 = 0 \Rightarrow x = \frac{768}{96} = 8 . ∴at x = 8, \frac{d^{2} p}{d x^{2}} = 96 \cdot > 0 So, x = 8 is a point of local minima So, y = 16 - 8 = 8 Hence, x = 8, y = 8 Q90. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible. A.17. Side of the tin square piece = 18 cm Let x cm be the thought of the square to be cut from each corner. The volume v of the box after cutting \Rightarrow v = length \times breadth \times height = (18 - 2 x)(18 - 2 x) x x = (18 - 2 x) 2 x = (324 + 4 x 2 - 72 x) x = 4 x 3 - 72 x 2 + 324 x So, \frac{d}{d x} = 12 x^{2} - 144 x + 324 \frac{d^{2} v}{d x^{2}} = 24 x - 144 As \frac{d}{d x} = 0 \to 12 x^{2} - 144 x + 324 = 0 \Rightarrow x 2 - 12 x + 27 = 0 \Rightarrow x 2 - 9 x - 3 x + 27 = 0 \Rightarrow x (x - 9) - 3 (x - 9) = 0 \Rightarrow (x - 9)(x - 3) = 0 \Rightarrow x = 9 and x = 3 At x = 0, length of box = 18 - 2π9 = 18 - 18 = 0 Which is not possible And at x = 3, \frac{d^{2} y}{d x^{2}} = 24(3) - 144 = -72 < 0 ∴x = 3 is a point of maximum Hence, ‘3’ cm (square) is to be cut from each side of the square So that volume of box is maximum Q91. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum? A.18. Let ‘x’ cm be the length of side of the square to be cut off from the rectangular surface Then, the volume v of the box is v = (45 - 2 x)(24 - 2 x) x = 1080 x - 138 x 2 + 4 x 3 So, \frac{d v}{d x} = 1080 - 276 x + 12 x^{2} \frac{d^{2}}{d x^{2}} = - 276 + 24 x At, \frac{d v}{d x} = 0 \to 1080 - 276 x + 12 x^{2} = 0 \Rightarrow x 2 - 23 x + 90 = 0 \Rightarrow x 2 - 5 x - 18 x + 90 = 0 \Rightarrow x (x - 5) - 18 (x - 5) = 0 \Rightarrow (x - 5) (x - 18) = 0 \Rightarrow x = 5 and x = 18 At x = 18, breadth = 24 - 2(18) = 24 - 36 = -12 which is not possible At, x = 5, \frac{d^{2} v}{d x^{2}} = - 276 + 24 (5) = - 276 + 120 = - 150 < 0 Hence, x = 5 is the point of maximum So, ‘5’ cm length of square seeds to be cut from each corner of the secthgle Q92. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. A.19. Let A.B.C.D.be the square increased in a given fixed circle with radius x Let ‘x’ and ‘y’ be the length and breadth of the rectangle ∴x, y> 0 In ABC, right angle at B, x 2 + y 2 = (2 x) 2 \Rightarrow x 2 + y 2 = 4 x 2 Q93. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. A.20. Let r and h be the radius and height of the cylinder So, r, h > 0 The total surface area s is given by S = 2πr(h + r) = content . \Rightarrow \frac{S}{2 π} = r (h + r) = content = x (say) \Rightarrow h + r = \frac{u}{r} \Rightarrow h = \frac{k}{r} - r = \frac{k - r^{2}}{r .} Then, the volume v of the cylinder V = π r^{2} h = π r^{2} (\frac{k - r^{2}}{r}) = π r (x - x^{2}) = π (k x - π^{3}) So, \frac{d v}{d r} = π (x - 3 r^{2}) \frac{d^{2}}{d r^{2}} = - 6 π r . For maximum, \frac{d v}{d r} = 0 \Rightarrow π [x - 3 x^{2}) = 0 \Rightarrow r^{2} = \frac{x}{3} Q94. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area? A21. The volume v of a cylinder of height h and radius r is V = π r 2 h = 100 \Rightarrow h = \frac{100}{π r^{2}} . Let, s be the surface area then S = 2 r 2 h r(r + h) = 2 x (r^{2} + \frac{100 r}{π r^{2}}) = 2 π (n^{2} + \frac{100}{π n}) . ∴ \frac{d S}{d r} = 2 π [2 x - \frac{100}{1 r^{2}}] \Rightarrow \frac{d^{2} S}{d r^{2}} = 2 π [2 + \frac{200}{π r^{3}}] At, \frac{d s}{d r} = 2 π [2 r - \frac{100}{π r^{2}}] = 0 \Rightarrow 2 r = \frac{100}{π r^{2}} \RightarrowM^{3} = \frac{50}{π} \Rightarrow M = {[\frac{50}{π}]}^{\frac{1}{3}}, > 0 . At, x = {[\frac{50}{π}]}^{\frac{1}{3}}, \frac{d^{2S}}{d r^{2}} = 2 π [2 + \frac{200}{π {[\frac{50}{π}]}^{\frac{1}{3}}} \times 3] = 2 π [2 + 4] = 12 π > 0 ∴ r = {[\frac{50}{π}]}^{\frac{1}{3}} isa point of minimum And h = \frac{100}{π {[\frac{50}{π}]}^{\frac{2}{3}}} = 2 {[\frac{50}{π}]}^{\frac{1}{3}} . Q95. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? A.22. Let x and y in ‘m’ be the length of side of the square the radius of the circle respectily Then, length of wire = perimeter of square + circumference of circle \Rightarrow 28 = 4x + 2πy \Rightarrow 2x + πy = 14 \Rightarrow y = \frac{14 - 2 x}{x} The combine area A of the square and the circle is A = x 2 + π y 2 = x^{2} + π {[\frac{14 - 2 x}{π}]}^{2} {= x^{2} + \frac{[2 (7 - x)}{x}]}^{2} = x^{2} + \frac{4}{π} {(7 - x)}^{2} . So, \frac{dA}{d x} = 2 x + \frac{4 \cdot 2}{π} (7 - x) + (- 1) = 2 x - \frac{8}{π} (7 - x) \frac{d^{2A}}{d x^{2}} = 2 + \frac{8}{π} At, \frac{dA}{d x} = 0 \Rightarrow 2 x - \frac{8 (7 - x)}{π} = 0 \Rightarrow 2 x = \frac{8}{π} (7 - x) \Rightarrow 2 π x = 56 - 8 x \Rightarrow (2 x + 8) x = 56 \Rightarrow x = \frac{56}{2 π + 8} = \frac{28}{x + 4} . At, x = \frac{28}{π + 4}, \frac{d^{2A}}{d x^{2}} = 2 + \frac{γ}{π} > 0 ∴ x = \frac{28}{π + 4} isa point of minima Hence, length of square = 4 x = 4 (\frac{28}{1 + 4}) = \frac{112}{π + 4} m and length of circle = 2πy = 2 π (\frac{14 - 2 x}{π}) = 2 π [\frac{14}{π} - \frac{2}{1} [\frac{28}{π + 4}]] = 28 - 4 [\frac{28}{π + 4}] = 28 - \frac{112}{1 + 4} = \frac{28 x + 112 - 112}{x + 4} = \frac{28 π}{π + 4} Q96. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \frac{8}{27} of the volume of the sphere. A.23. Let r and h be the radius and height of the one in scribed in the sphere of radius R. Then, is ΔOBC, rt angle at B (h - r) 2 + r 2 = R 2 \Rightarrow h 2 + R 2 - 2 h R + h 2 = R 2 \Rightarrow r 2 = 2 h R - h 2 Then the volume v of the cone is, V = \frac{1}{3} \times r^{2} h = \frac{1}{3} π (2 h R - h^{2}) h = \frac{1}{3} π (2 {Rh}^{2} - h^{3}) . \Rightarrow \frac{dV}{d h} = \frac{π}{3} (4 Rh - 3 h^{2}) . \frac{d^{2V}}{d h^{2}} = \frac{π}{3} (4R - 6 h) . At \frac{dV}{d h} = 0 - \frac{π}{3} (4R h - 3 h^{2}) = 0 . 4Rh - 3 h 2 = 0. h(4R - 3h) = 0. h = 0 and h = \frac{4 R}{3} As h> 0, h = \frac{4}{3} . At h = \frac{4R}{3}, \frac{d^{2v}}{d h^{2}} = \frac{π}{3} [4 R - 6 \times \frac{4rR}{3}] = \frac{1}{3} [4R - 8 R] = - \frac{4 Rπ}{3} < 0 h = \frac{4R}{3} is a point of maxima. and r^{2} = 2 hR - h^{2} = 2 \times \frac{4R}{3} R- {(\frac{4R}{3})}^{2} = \frac{3 \times 8R^{2}}{9} - \frac{1 6R^{2}}{9} r^{2} = \frac{8R^{2}}{9} Hence, Volume of Cone, = \frac{1}{3} π r^{2} h = \frac{1}{3} π \times \frac{8R^{2}}{9} \times \frac{4R}{3} = \frac{8}{27} \times \frac{4}{3} πR^{3} = \frac{8}{27} \times Volume of sphere. Q97. Show that the right circular cone of least curved surface and given volume has an altitude equal to \sqrt2 time the radius of the base. A.24. Let r and h be the radius and height of the cone. The volume V of the cone is. = \frac{1}{3} π r^{2} h r^{2} h = \frac{3V}{π} = k(SAY) \Rightarrow r^{2} = \frac{k}{h} . And curve surface area S is Q98. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is t a n^{- 1} \sqrt2. A.25. Let r, h, l and Ø be the radius, height, slant height and semi-vertical angle respectively of the cone. i.e., r, h, l>0. Then, Volume V of the cone is = \frac{1}{3} π r^{2} h . = \frac{1}{3} π (l^{2} - h^{2}) h = \frac{1}{3} π (l^{2} h - h^{3}) . So, \frac{dV}{d h} = \frac{1}{3} π (l^{2} - 3 h^{2}) \frac{d^{2V}}{d h^{2}} = - 2 π h . Q99. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is s i n^{- 1} \frac{1}{3} . A.26. Choose the correct answer in Questions 27 and 29. Q100. The point on the curve x2 = 2y which is nearest to the point (0, 5) is (A) (2\sqrt2,4) (B) (2\sqrt2,0) (C) (0, 0) (D) (2, 2) A.27. The equation of the given curve is x 2 = 2y. Let p(x, y) be a point on the curve. The distance of p(x, y) from (0, 5) is say S is given by s^{2} = x^{2} + {(y - 5)}^{2} Let z = s 2 = x 2 + y 2 + 25 - 10 y = 2 y + y 2 -10 y + 25 z = y 2 - 8 y + 25 So, \frac{d z}{d y} = 2 y - 8 \frac{d^{2} z}{d y^{2}} = 2 At \frac{d z}{d y} = 0 \Rightarrow 2 y - 8 = 0 \Rightarrow y = \frac{8}{2} = 4 At y = 4, \frac{d^{2} z}{d y^{2}} = 2 > 0 y = 4, is point of minimum distance. So, x 2 = 2 y --> x 2 = 2 \times 4--> x 2 = 8 x = \pm 2 Hence, the point of nearest distance are (2\sqrt2, 4) and (- 2\sqrt2, 4) . Option (A) is correct. Q101. For all real values of x, the minimum value of \frac{1 - x + x^{2}}{1 + x + x^{2}} is (A) 0 (B) 1 (C) 3 (D) \frac{1}{3} A.28. We have, f (x) = \frac{1 - x + x^{2}}{1 + x + x^{2}} . f (x) = \frac{(1 + x + x^{2}) \frac{d}{d x} (1 - x + x^{2}) - (1 - x + x^{2}) \frac{d}{d x} (1 + x + 2)}{{(1 + x + x^{2})}^{2}} \Rightarrow f^{'} (x) = \frac{(1 + x + x^{2}) (- 1 + 2 x) - (1 - x + x^{2}) (1 + 2 x)}{{(1 + x + x^{2})}^{2}} = \frac{- 1 + 2 x - x + 2 x^{2} - x^{2} + 2 x^{3} - 1 - 2 x + x + 2 x^{2} - x^{2} - 2 x^{3}}{{(1 + x + x^{2})}^{2}} = \frac{- 2 + 2 x^{2}}{{(1 + x + x^{2})}^{2}} = \frac{- 2 (1 - x^{2})}{{(1 + x + x^{2})}^{2}} = - \frac{2 (1 + x) (1 - x)}{{(1 + x + x^{2})}^{2}} At f(x) = 0. \Rightarrow \frac{- 2 (1 + x) (1 - x)}{{(1 + x + x^{2})}^{2}} = 0 x = 1 and x = -1. At x = 1, f (1) = \frac{1 - 1 + 1}{1 + 1 + 1} = \frac{1}{3} x = 1, f (1) = \frac{1 - 1 + 1}{1 + 1 + 1} = \frac{1}{3} At x = -1, f (- 1) = \frac{1 + 1 + 1}{1 - 1 + 1} = 3 The maximum value of f(x) = \frac{1}{3} x = 1 Hence, option (D) is correct. Q102. The maximum value of {[x (x - 1) + 1]}^{\frac{1}{3}}, 0 (A) {\frac{1}{3}}^{\frac{1}{3}} (B) \frac{1}{2} (C) 1 (D) 0 A.29. We have, f^{'} (x) = [X(x - 1) {+ 1]}^{\frac{1}{3}}, 0 \leq x \leq 1 f (x) = \frac{1}{3} {[x^{2} - x + 1]}^{\frac{- 2}{3}} \frac{d}{d x} (x^{2} - x + 1) = \frac{2 x - 1}{3 {(x^{2} - x + 1)}^{\frac{2}{3}}} At f(x) = 0. 2x - 1 = 0 x = \frac{1}{2} \in [0, 1] f (\frac{1}{2}) = {[\frac{1}{2} (\frac{1}{2} - 1) + 1]}^{\frac{1}{3}} = {[\frac{1}{4} - \frac{1}{2} + 1]}^{\frac{1}{3}} . = {[\frac{1 - 2 + 4}{4}]}^{\frac{1}{3}} = {[\frac{3}{4}]}^{\frac{1}{3}} ≃ 0.90 f (0) = [0(0 - 1) + {1]}^{\frac{1}{3}} = 1 . f (1) = [1 (1 - 1) {+ 1]}^{\frac{1}{3}} = 1 . Option (B) is Hence maximum value of f(x) = at x = 1 and x = 0. Option (c) is correct.NCERT Class 12 Application of Derivatives Miscellaneous Exercise Solutions The Class 12 Application of Derivatives Miscellaneous Exercise contains problems based on a range of concepts covered in the chapter, including the rate of change, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, and optimization problems. This exercise is designed to test students' understanding and ability to apply these concepts in a variety of challenging situations. Miscellaneous Exercise Solutions will have answers to 24 Questions including 14 Long, 4 Short, and 6 MCQs. Students can check the complete solutions of the Chapter 6 Miscellaneous Exercise below; Application of Derivatives Miscellaneous Exercise Solution Q103. Using differentials, find the approximate value of each of the following (a) {(\frac{17}{81})}^{\frac{1}{4}} A.1. (a) consider y = x^{\frac{1}{4}} . L e t, x = \frac{16}{81} & △ x = \frac{1}{81} . \begin{array}{l} T h e n, △ y = {(x + △ x)}^{\frac{1}{4}} - x^{\frac{1}{4}} \\ = {(\frac{17}{81})}^{\frac{1}{4}} - {(\frac{16}{81})}^{\frac{1}{4}} \\ = {(\frac{17}{81})}^{\frac{1}{4}} - \frac{2}{3} \\ ∴ {(\frac{17}{81})}^{\frac{1}{4}} = \frac{2}{3} + △ y \end{array} Now, d y is approximately equal to △ y and is given by, \begin{array}{l} d y = (\frac{d y}{d x}) △ x = \frac{1}{4 {(x)}^{\frac{3}{4}}} (△ x) (a s, y = x^{\frac{1}{4}}) \\ = \frac{1}{4 {(\frac{16}{81})}^{\frac{3}{4}}} (\frac{1}{81}) = \frac{27}{4 \times 8} \times \frac{1}{81} = \frac{1}{32 \times 3} = \frac{1}{96} = 0.010 \end{array} Hence, the approximate value of {(\frac{17}{81})}^{\frac{1}{4}} is \frac{2}{3} + 0.010 = 0.667 + 0.010 =0.677 (b) {(33)}^{\frac{- 1}{5}} (b) Consider y = x^{\frac{1}{5}} . L e t, x = 32 & △ x = 1 Now, d y is approximately equal to △ y and is given by, \begin{array}{l} d y = (\frac{d y}{d x}) (△ x) = \frac{- 1}{5 {(x)}^{\frac{6}{5}}} (△ x) (a s, y = x^{\frac{1}{5}}) \\ = - \frac{- 1}{5 {(x)}^{6}} (1) = - \frac{1}{320} = - 0.003 \end{array} Hence, the approximate value of \begin{array}{l} 33^{\frac{1}{5}} \end{array} is \frac{1}{2} + (- 0.003 = 0.497) Q104. Show that the function given by f (x) \frac{l o g x}{x}, has maximum at x = e. A2. We have, f(x) \frac{l o g x}{x}, x > 0 . f(x) = \begin{array}{l} \underline{x \frac{d}{d x} l o g x - l o g x \frac{d}{d x} x} \\ x^{2} . \end{array} = \frac{x \times \frac{1}{x} - l o g x}{x^{2}} = \frac{1 - l o g x}{x^{2}} f(x) = \frac{x^{2} \frac{d}{d x} (1 - l o g x) - (1 - l o g x) \frac{d}{d x} x^{2}}{x^{4}} = \frac{x^{2} (- \frac{1}{x}) - (1 - l o g x) (2 x)}{x^{4}} = \frac{- 1 - 2 + 2 l o g x}{x^{3}} = \frac{2 l o g x - 3}{x^{3}} At extreme points, f(x) = 0. \Rightarrow \frac{1 - l o g x}{x^{2}} = 0 . \Rightarrow l o g x = 1 = l o g e . \Rightarrow x = e . At x = e, f"(e) = \frac{2 l o g e - 3}{e^{3}} = \frac{2 - 3}{e^{3}} = - \frac{1}{e^{3}} < 0 x = e is a point of maximum. Q105. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ? A3. Let ‘b’ and ‘x’ be the fixed base and equal side of isosceales triangle. Then, \frac{d x}{d t} = - 3 cm/s (Ø decreasing). Q106. Find the equation of the normal to curve x 2 = 4y which passes through the point (1, 2) A4. Equation of the curve is y^{2} = 4 x . . . . . . . . . . (i) \begin{array}{l} 2 y \frac{d y}{d x} = 4 \\ \Rightarrow \frac{d y}{d x} = \frac{4}{2 y} = \frac{2}{y} \\ ∴ {\frac{d y}{d x}]}_{(1, 2)} = \frac{2}{2} = 1 \end{array} Now, the slope of the normal at point (1, 2) is \frac{- 1}{{\frac{d y}{d x}]}_{(1, 2)}} = \frac{- 1}{1} = 1 ∴ Equation of the normal at (1, 2) is y - 2 = - 1 (x - 1) . \begin{array}{l} \Rightarrow y - 2 = - x + 1 \\ \Rightarrow x + y - 3 = 0 \end{array} Q107. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ - aθ cosθ is at a constant distance from the origin. A5. We have x = a c o s θ + a θ s i n θ \begin{array}{l} ∴ \frac{d x}{d θ} = - a s i n θ + a s i n θ + a θ c o s θ = a θ c o s θ \\ y = a s i n θ - a θ c o s θ \\ ∴ \frac{d y}{d θ} = a c o s θ - a c o s θ + a θ s i n θ = a θ s i n θ \\ ∴ \frac{d y}{d x} = \frac{d y}{d θ} . \frac{d θ}{d x} = \frac{a θ s i n θ}{a θ c o s θ} = t a n θ \end{array} ∴ Slope of the normal at any point θ is - \frac{1}{t a n θ} The equation of the normal at a given point (x, y) is given by, \begin{array}{l} y - a s i n θ + a θ c o s θ = \frac{- 1}{t a n θ} (x - a c o s θ - a θ s i n θ) \\ \Rightarrow y s i n θ - a s i n^{2} θ + a θ s i n θ c o s θ = - x c o s θ + a c o s^{2} θ + a θ s i n θ c o s θ \\ \Rightarrow x c o s θ + y s i n θ - a (s i n^{2} θ + c o s^{2} θ) = 0 \\ \Rightarrow x c o s θ + y s i n θ - a = 0 \end{array} Now, the perpendicular distance of the normal from the origin is Q108. Find the intervals in which the function f given by f(x)= \frac{4 s i n x - 2 x - x c o s x .}{2 + c o s x} is (i) increasing (ii) decreasing A6. We have, f(x)= \frac{4 s i n x - 2 x - x c o s x .}{2 + c o s x} = \frac{4 s i n x - x (2 + c o s x)}{2 + c o s x} f (x) = \frac{4 s i n x}{2 + c o s x} - x . So, f^{'} (x) = \frac{(2 + c o s x) \frac{d}{d x} (4 s i n x) - 4 s i n x \frac{d}{d x} (2 + c o s x)}{{(2 + c o s x)}^{2}} - \frac{d x}{d x} . = \frac{(2 + c o s x) (4 c o s x) - (4 s i n x) (- s i n x)}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 c o s x + 4 c o s^{2} x + 4 s i n^{2} x}{{(2 + c o s x)}^{2}} - 1 = \frac{8 c o s x + 4 (c o s^{2} x + s i n^{2} x)}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 c o s x + 4}{{(2 + c o s x)}^{2}} - 1 . = \frac{8 \cdot c o s x + 4 - {(2 + c o s x)}^{2}}{{(2 + c o s x)}^{2}} = \frac{8 c o s x + 4 - 4 - 4 c o s x - c o s^{2} x}{{(2 + c o s x)}^{2}} = \frac{4 c o s x - c o s^{2} x}{{(2 + c o s x)}^{2}} = \frac{c o s x (4 - c o s x)}{{(2 + c o s x)}^{2}} . Now, {(2 + c o s x)}^{2} > 0 . And, 4 - c o s x > 0 as cos x lies in [1, 1]. So, (i) for increasing, f(x) \geq 0. cosx \geq 0. x lies in Ist and IVth quadrant. i.e., f(x) is increasing for 0 \leq x \leq \frac{x}{2} and \frac{3 x}{2} \leq x \leq 2 x . (ii) for decreasing, f(x) \leq 0. cosx \leq 0. x lies in IInd and IIIrd quadrant. i.e., f(x) is decreasing for \frac{x}{2} \leq x \leq \frac{3 x}{2} . Q109. Find the intervals in which the function f given by f (x) = x^{3} + \frac{1}{x^{3}}, x \neq 0 is (i) Increasing (ii) Decreasing A7. We have, f(x) = x^{3} + \frac{1}{x^{3}}, x \neq 0 . \Rightarrow f^{'} (x) = 3 x^{2} - \frac{3 \times 1}{x^{4}} = 3 (x^{2} - \frac{1}{x^{4}}) = 3 (\frac{x^{6} - 1}{x^{4}}) . = \frac{3}{x^{4}} [{(x^{2})}^{3} - 1^{3}] = \frac{3}{x^{4}} (x^{2} - 1) (x^{4} + x^{2} + 1) {(a^{3} - b^{3}) = (a - b) (a^{2} + b^{2} + a b) \Rightarrow f^{'} (x) = \frac{3 (x - 1)}{x^{4}} (x + 1) (x^{4} + x^{2} + 1) . {∵ x^{2} - b^{2} = (a - b) (a + b) At f^{'} (x) = 0 . \Rightarrow \frac{3 (x - 1) (x + 1) (x^{4} + x^{2} + 1)}{x^{4}} = 0 \Rightarrow x = 1 x = - 1 . 3 (x^{4} + x^{2} + 1) \neq 0 . So we have three disjoint internal i.e., (- \infty, - 1], [- 1, 1] (1, \infty) . When, x \in (- \infty, - 1] . f^{'} (x) = \frac{(+) v e (- v e) (- v e) (+ v e)}{(+ v e)} = (+) v e and 0 at x = - 1 . So, f(x) is increasing. When x \in [- 1, 1] f^{'} (x) = (+ v e) (- v) (+ v e) = (- v e) 0 at x = 1and - 1 So, f(x) is decreasing. When x \in [1, \infty) f(x) = (+ v e) (+ v e) (+ v e) = (+ v e) on 0 at x = 1 So, f(x) is increasing. f(x) is increasing for x \in (\infty,1) and [1, \infty] and decreasing for x \in [1, 1]. Q110. Find the maximum area of an isosceles triangle inscribed in the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 with its vertex at one end of the major axis. A.8. The given equation of the ellipse is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (1) Let the major axis be along x-axis so, vertex is at (\pm a, 0) Let ΔABC be the isosceles triangle inscribed on the ellipse with one vertex C at (a, 0). Then, let A have Co-ordinate (x0, yo) from figure. So, Co-ordinate of B = (x0, y0) As A and B lies on the ellipse, from equation (i), Q111. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m 3 . If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank? A.9. Let x and y meters be the length and breath of the rectangular base of the tank respectively. Then, volume V of the tank is V = length \times depth \times breath. V = 2xy = 8m 3 (given). \Rightarrow y = \frac{8}{2 x} = \frac{4}{x} . Let ‘t’ be the total cost of building the tank. Then, t = cost of base + cost of sides. = 70xy + 45[4x+4y] {there are four sides. = 70xy + 180x+ 180y. = = 70 \cdot x \cdot \frac{4}{x} + 180 x + 180 \times \frac{4}{x} . t = 280 + 180 x + \frac{720}{x} . So, \frac{d z}{d x} = 0 + 180 - \frac{720}{x^{2}} . And \frac{d^{2} t}{d x^{2}} = \frac{1440}{x^{3}} At \frac{d^{2}}{d x} = 0 \Rightarrow 180 - \frac{720}{x^{2}} = 0 \Rightarrow x^{2} = \frac{720}{180} = 4 x = \pm 2 x = 2, (x = length and it cannot be negative) At x = 2, \frac{d^{2} t}{d x^{2}} = \frac{1440}{2^{3}} = \frac{1440}{8} = 180 > 0 . x = 2 is point of maxima. Hence, minimum cost = 280 + 180 \times 2 + \frac{720}{2} = 280 + 360 + 360 = 1000. Q112. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. A.10. Let r and s be the radius of the circle and length of side of square. Then, sum of perimeter of circle and square = k 2πr +4s = k s = \frac{k - 2 \times r}{4} The area A be the total areas of the circle and square. Then, A = π r 2 + s 2 = π r^{2} + {(\frac{k - 2 π r}{4})}^{2} = π r^{2} + \frac{x^{2} + 4 π^{2} r^{2} - 4 π r k}{16} = \frac{16 π r^{2} + x^{2} + 4 π^{2} r^{2} - 4 π k \cdot r}{16} = \frac{1}{16} [(16 π + 4 π^{2}) π^{2} - 4 π k r + x^{2}] . So, \frac{d A}{d r} = \frac{1}{16} [(16 π + 4 x^{2}) \cdot 2 n - 4 π k .] . And \frac{d^{2A}}{d r^{2}} = \frac{1}{16} [(6 π + 4 π^{2}) 2] = \frac{16 π + 4 π^{2}}{8} At \frac{d A}{d r} = 0 \frac{1}{16} [(16 x + 4 x^{2}) 2 x - 4 π k] = 0 \Rightarrow (16 x + 4 x^{2}) 2 x - 4 π k = 0 \Rightarrow 4 π [4 + π] 2 r = 4 π k . r = \frac{k}{2 (4 + π) .} At r = \frac{x}{2 (4 + π)}, \frac{d^{2A}}{d x^{2}} = \frac{16 π + 4 π^{2}}{8} > 0 . s = \frac{2 x}{2 (4 + x) .} s = 2r. Hence, proved. Q113. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. A.11. Let ‘x’ metre be the radius of the semi-circular opening mounded on the length ‘2x’ side of rectangle. Then, let ‘y’ be the breadth of the rectangle. Then, perimeter of the window = 10m \Rightarrow \frac{2 π x}{2} + 2 x + y + y = 10 x + 2x + 2y + = 10. \Rightarrow y = \frac{10 - (π + 2) π}{2 .} Let the area of the window be A. Then, A = \frac{1}{2} (π x^{2}) + 2 x y . = \frac{π x^{2}}{2} + 2 x \cdot [\frac{10 - (x + 2) x}{2}] . = \frac{π x^{2} + 20 x - 2 π x^{2} - 4 x^{2}}{2} = \frac{1}{2} [-π x 2 - 4 x 2 + 20 x]. So, \frac{d}{d x} = \frac{1}{2} [ -2πx - 8x + 20] And \frac{d \overset{2}{}}{d x^{2}} = \frac{1}{2} [ -2π - 8] = -π -4 = -( π+ 4) At \frac{d}{d x} = 0 . \frac{1}{2} [ -2πx - 8x + 20] = 0 2x + 8x = 20 x = \frac{20}{2 π + 8} = \frac{10}{π + 4} . At x = \frac{10}{π + 4}, \frac{d^{2A}}{d x^{2}} = -( π+ 4) < 0 Øx = \frac{10}{x + 4} is a point of minima. And y = \frac{10 - (x + 2) \times (\frac{10}{x + 4})}{2} = \frac{10 (π + 4) - (π + 2) 10}{2 (π + 4)} = \frac{10 π + 40 - 10 π - 20}{2 (x + 4)} = \frac{10}{π + 4 .} Ø Dimensions of the window are length = 2x = \frac{20}{π + 4} m breadth = y \frac{10}{π + 4} m . radius = y = \frac{10}{π + 4} m . Q114. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is {(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{3}{2}} . A.12. Let P be the point on hypotenuse of a triangle. ABC, t angle at B. Which is at distance a& b from the sides of the triangle. Let < BAC = < MPC = Ø. Then, in\dots ΔANP, \Rightarrow t a n^{3} θ = \frac{a}{b} t a n θ = {(\frac{a}{b})}^{\frac{1}{3} .} At tan Ø = {(\frac{a}{b})}^{\frac{1}{3,}} \frac{d^{2} \dot{q}}{d θ^{2}} = + v e . > 0 . { Øall trigonometric f x n are + ve in I st quadrant}. So, z is least for tanØ = {(\frac{a}{b})}^{\frac{1}{3 .}} As, Sec 2 Ø = 1 + tan 2 Ø = 1 + {(\frac{a}{b})}^{\frac{2}{3 .}} = \frac{b^{\frac{2}{3}} + a^{\frac{2}{3}}}{b^{\frac{2}{3 .}}} secØ = {[\frac{b^{\frac{2}{3}} + a^{\frac{2}{3}}}{b^{\frac{2}{3}}}]}^{\frac{1}{2}} = \frac{{(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{1}{2}}}{b^{\frac{1}{3} .}} And tan 2 Ø = \frac{1}{c o t^{2} θ} cot 2 Ø = {(\frac{a}{b})}^{\frac{2}{3}} And cosec 2 Ø = 1 + cot 2 Ø = 1 + {(\frac{a}{b})}^{\frac{2}{3}} = \frac{a^{\frac{2}{3}} + b^{\frac{2}{3}}}{a^{\frac{2}{3}}} cosecØ = \frac{{(a^{\frac{2}{3}} + b^{\frac{2}{3}})}^{\frac{1}{2} .}}{a^{\frac{1}{3}}} Q115. Find the points at which the function f given by f (x) = (x - 2) 4 (x + 1) 3 has (i) local maxima (ii) local minima (iii) point of inflexion A.13. We have, f(x) = (x - 2) 4 (x + 1) 3 . So, f(x) = (x - 2) 4 . 3(x + 1) 2 + (x + 1) 3 . 4(x - 2) 3 . = (x - 2) 3 [x + 1) 2 [3 (x - 2) + 4(x + 1)] = (x - 2) 3 (x + 1) 2 (3 x - 6 + 4 x + 4) = (x - 2) 3 (x + 1) 2 (7 x - 2). At f(x) = 0. (x - 2) 3 (x + 1) 2 . (7 x - 2) = 0. x = 2, x = -1 or x = \frac{2}{7} . As (x + 1) 2 > 0,we shave evaluate for the remaining factor. At x = 2, When x< 2, f(x) = ( -ve)(+ ve)(+ ve) = ( -ve) < 0. When x> 2, f(x) = (+ ve)(+ ve)(+ ve) = (+ ve) > 0. Øf(x) change from ( -ve) to (+ ve) as x increases So, x = 2 is a point of local minima At x = -1. When x< -1, f(x) = ( -ve)(+ ve)( -ve) =\notin , ve > 0. When x> -1, f(x) = ( -ve)(+ ve)(+ ve) =\notin , ve > 0. So, f(x) does not change through x ---1. Hence, x = -1 is a point of infixion At x = \frac{2}{7,} When x< \frac{2}{7}, f(x) = ( -ve)(+ ve)( -ve) = ( -ve > 0. When x< \frac{2}{7}, f(x) = ( -ve)(+ ve)(+ ve) = ( -ve < 0. f(x) change from (+ ve) to ( -ve) as x increases through \frac{2}{7} . x = \frac{2}{7} is a point of local maxima Q116. Find the absolute maximum and minimum values of the function f given by f (x) =cos 2 x + sin x, x \in [0, π] A.14. We have, f(x) = cos 2 x + sin x, x\in [0,π ]. So, f(x) = 2 cos x ( -sin x) + cos x = cos x (1 - 2 sin x). At f(x) = 0 cosx (1 - 2 sin x) = 0 cosx = 0 or 1 - 2 sin x = 0 cosx = cos \frac{π}{2} or sin x = \frac{1}{2} = sin \frac{π}{6} = sin x - \frac{π}{6} x= \frac{π}{2}, x = \frac{π}{6} and x = \frac{5 π}{6} [0,π ]. So, f (\frac{π}{2}) = cos 2 \frac{π}{2} + sin \frac{π}{2} = 1. Absolute minimum of f(x) = \frac{5}{4} and absolute minimum of f(x) = 1. Q117. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \frac{4 r}{3} . A.15. Let x be the radius of the sphere ad x be radius of the right circular cone. Let height of cone = y Then, in ΔOBA, (y - r) 2 + x 2 = r 2 y 2 + r 2 - 2 ry + x 2 = r 2 x 2 =2 ry - y 2 So, the volume V of the cone is V= \frac{1}{3} π \cdot x^{2} \cdot y {= \frac{1}{3} π {(r a d i u s)}^{2} \timeshight} = \frac{1}{3} \times (2 x y - y^{2}) y = \frac{1}{3} x (2 x y^{2} - y^{3}) . So, \frac{d v}{d y} = \frac{π}{3} [4 x y - 3 y^{2}] And \frac{d^{2}}{d y^{2}} = \frac{x}{3} [4 x - 6 y] . At \frac{d V}{d y} = 0 . \Rightarrow \frac{π}{3} [4 x y - 3 y^{2}] = 0 4 x - y - 3 y 2 = 0 \Rightarrow y = \frac{4 x}{3} asy> 0. At y = \frac{4 π}{3}, \frac{d^{2} v}{d y^{2}} = \frac{π}{3} [4 x - 8 x] = - \frac{4 π r}{3} < 0 . Ø V is maximum when y = \frac{4 π}{3} Q.118. Let f be a function defined on [a, b] such that f '(x) > 0, for all x \in (a, b). Then prove that f is an increasing function on (a, b). A.16. We have, f(x) defined on [a, b] And f(x) > 0 \forall x \in [a, b]. Let x 1, x 2 \in [a, b] and x 2 > x 1 In the internal x 1, x 2], f(x) will also be continuous and differentiable. Hence by mean value theorem, there exist c [x 1, x 2] such that f (c) = \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1 .}} f(x) > 0 \forall x \in [a, b]. Then, f(c) > 0. \Rightarrow \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1}} > 0 i.e., f(x 2) - f (x 1) > 0 f(x 2) > f (x 1). Hence, the function f(x) is always increasing on [a, b] Q119. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/\sqrt3. Also find the maximum volume. Q120. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is \frac{4}{27} π h^{3} t a n^{2} α . A.18. Choose the correct answer in the questions from 19 to 24. Q121. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h A.19. Let x be the depth of the wheat inside the cylindrical tank is with radius = 10 cm Then, Volume V of the cylindrical tank is V = π(10) 2 x = 100πp x m 3 As \frac{d}{d t} = 314 \frac{3}{h} \Rightarrow 100 π \frac{d x}{d t} = 314 \frac{d x}{d t} = \frac{314}{100 π} = \frac{314}{100 π 3.14} = \frac{314}{314} = 1 . \frac{m}{h} i e, rate of increasing of depth option (A) is correct Q122. The line y = mx + 1 is a tangent to the curve y 2 = 4x if the value of m is (A) 1 (B) 2 (C) 3 (D) \frac{1}{2} A.21. The equation of the tangent to the given curve is y = m x + 1 . Now, substituting y = m x + 1 . in y^{2} = 4 x, we get: \begin{array}{l} \Rightarrow {(m x + 1)}^{2} = 4 x \\ \Rightarrow m^{2} x^{2} + 1 + 2 m x - 4 x = 0 \\ \Rightarrow m^{2} x^{2} + x (2 m - 4) + 1 = 0 . . . . . . . . . . (i) \end{array} Since a tangent touches the curve at one point, the roots of equation (i) must be equal. Therefore, we have: Discriminant = 0 \begin{array}{l} {(2 m - 4)}^{2} - 4 {(m)}^{2} (1) = 0 \\ \Rightarrow 4 m^{2} + 16 - 16 m - 4 m^{2} = 0 \\ \Rightarrow 16 - 16 m = 0 \\ \Rightarrow m = 1 \end{array} Hence, the required value of m is 1. Therefore, option (A) is correct. Q123. The normal at the point (1,1) on the curve 2y + x 2 = 3 is (A) x + y = 0 (B) x - y = 0 (C) x + y +1 = 0 (D) x - y = 1 A.22. The equation of the given curve is 2 y + x^{2} = 3 . Differentiate with respect to x, we have: \begin{array}{l} \frac{2 d y}{d x} + 2 x = 0 \\ \Rightarrow \frac{d y}{d x} = - x \\ ∴ {\frac{d y}{d x}]}_{(1, 1)} = - 1 \end{array} The slope of the normal to the given curve at point (1,1) is \frac{- 1}{{\frac{d y}{d x}]}_{(1, 1)}} = - 1 Hence, the equation of the normal to the given curve at (1,1) is given as: \begin{array}{l} \Rightarrow y - 1 = 1 (x - 1) \\ \Rightarrow y - 1 = x - 1 \\ \Rightarrow x - y = 0 \end{array} Therefore, option (B) is correct. Q124. The normal to the curve x 2 = 4y passing (1,2) is (A) x + y = 3 (B) x - y = 3 (C) x + y = 1 (D) x - y = 1 A.23. The equation of the given curve is x^{2} = 4 y Differentiating with respect to x, we have: \begin{array}{l} 2 x = 4 . \frac{d y}{d x} \\ \Rightarrow \frac{d y}{d x} = \frac{x}{2} \end{array} The slope of the normal to the given curve at point (h,k) is given by, \frac{- 1}{{\frac{d y}{d x}]}_{(h, k)}} = - \frac{2}{h} ∴ Equation of the normal at point (h,k) is given as: y - k = \frac{- 2}{h} (x - h) Now, it is given that the normal passes through the point (1,2). Therefore, we have: 2 - k = \frac{- 2}{h} (1 - h) o r, k = 2 + \frac{2}{h} (1 - h) . . . . . . . . . . (i) Since (h,k) lies on the curve x^{2} = 4 y,we have h^{2} = 4 k \Rightarrow k = \frac{h^{2}}{4} From equation (i), we have: \begin{array}{l} \frac{h^{2}}{4} = 2 + \frac{2}{h} (1 - h) \\ \Rightarrow \frac{h^{3}}{4} = 2 h + 2 - 2 h = 2 \end{array} \begin{array}{l} \Rightarrow h^{3} = 8 \\ \Rightarrow h = 2 \\ ∴ k = \frac{h^{2}}{4} \Rightarrow k = 1 \end{array} Hence, the equation of the normal is given as: \begin{array}{l} \Rightarrow y - 1 = \frac{- 2}{2} (x - 2) \\ \Rightarrow y - 1 = - (x - 2) \\ \Rightarrow x + y = 3 \end{array} Therefore, option (A) is correct. Q125. The points on the curve 9y 2 = x 3, where the normal to the curve makes equal intercepts with the axes are (A) (4, \pm \frac{8}{3}) (B) 4, \frac{- 8}{3} (C) 4, \pm \frac{3}{8} (D) \pm 4, \frac{3}{8} A.24. The equation of the given curve is 9 y^{2} = x^{3} Differentiate with respect to x, we have: \begin{array}{l} 9 (2 y) \frac{d y}{d x} = 3 x^{2} \\ \Rightarrow \frac{d y}{d x} = \frac{x^{2}}{6 y} \end{array} The slope of the normal to the given curve at point (x_{1}, y_{1}) is \frac{- 1}{{\frac{d y}{d x}]}_{(x_{1}, y_{1})}} = - \frac{6 y_{1}}{{x_{1}}^{2}} ∴ The equation of the normal to the curve at (x_{1}, y_{1}) is \begin{array}{l} y - y_{1} = - \frac{6 y_{1}}{{x_{1}}^{2}} (x - x_{1}) \\ \Rightarrow {x_{1}}^{2} y - {x_{1}}^{2} y_{1} = - 6 x y_{1} + 6 x_{1} y_{1} \\ \Rightarrow 6 x y_{1} + {x_{1}}^{2} y = 6 x_{1} y_{1} + {x_{1}}^{2} y_{1} \\ \Rightarrow \frac{6 x y_{1}}{6 x_{1} y_{1} + {x_{1}}^{2} y_{1}} + \frac{{x_{1}}^{2} y}{6 x_{1} y_{1} + {x_{1}}^{2} y_{1}} = 1 \\ \Rightarrow \frac{x}{\frac{x_{1} (6 + x_{1})}{6}} + \frac{y}{\frac{y_{1} (6 + x_{1})}{x_{1}}} \end{array} It is given that the normal makes intercepts with the axes. Therefore, we have: \begin{array}{l} ∴ \frac{x_{1} (6 + x_{1})}{6} = \frac{y_{1} (6 + x_{1})}{x_{1}} \\ \Rightarrow \frac{x_{1}}{6} = \frac{y_{1}}{x_{1}} \\ \Rightarrow {x_{1}}^{2} = 6 y_{1} . . . . . . . . . . (i) \end{array} Also, the point (x_{1}, y_{1}) lies on the curve, so we have 9 {y_{1}}^{2} = {x_{1}}^{3} . . . . . . . . . . (i i) From (i) and (ii), we have: 9 (\frac{{x_{1}}^{2}}{6}) = {x_{1}}^{3} \Rightarrow \frac{{x_{1}}^{4}}{4} = {x_{1}}^{3} \Rightarrow x_{1} = 4 From (ii), we have: \begin{array}{l} 9 {y_{1}}^{2} = {(4)}^{3} = 64 \\ \Rightarrow {y_{1}}^{2} = \frac{64}{9} \\ \Rightarrow y_{1} = \pm \frac{8}{3} \end{array} Hence, the required points are (4, \pm \frac{8}{3}) . Therefore, option (A) is correct. Read moreMaths Ncert Solutions class 12th Exam Student Forum Anything you would want to ask experts? 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tangent and normal to a curve at a point, to determine the rate of change of quantities, and to find turning points on the graph of a function. The chapter covers the applications of the derivative in various disciplines such as Science, Engineering, Social Sciences, and other fields.\u003cbr>\u003cstrong>Application of Derivatives Class 12 NCERT Solutions\u003c/strong> covers the increasing and decreasing functions, the rate of change of quantities, and maxima and minima. The subject matter experts at Shiksha have created the NCERT solution to help students deepen their understanding of concepts. It is ideal for exam preparation. The detailed and step-by-step solutions help students to score high in the \u003cstrong>CBSE Board exam\u003c/strong> and competitive exams like \u003cstrong>JEE Mains\u003c/strong>.\u003cbr>To get the NCERT Solutions of Class 11 and 12 of all three subjects, including Chemistry, Maths, and Physics, check - \u003ca href=\"https://www.shiksha.com/preparation/ncert-solutions-ns\">NCERT Solutions Class 11 and 12\u003c/a>.\u003c/p>","order":1,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"intro","wikiIdentifier":"106185","status":"live","updateDateOn":false,"updatedOn":"2025-05-16 18:06:24","widgetData":[{"dataValue":"LIVE_BLOG_WIDGET","order":1,"dataCategoryType":"widget","updateDateOn":false}]},{"id":952240,"h2Text":"Quick Overview of Applications of Derivatives - Class 12 Maths","dataValue":"\u003cp>Here is a summary of the \u003cstrong>Applications of Derivatives Class 12 NCERT Solutions\u003c/strong>:\u003c/p> \n\u003cul> \n \u003cli>The chapter covers the rate of change of quantities and the chain rule.\u003c/li> \n \u003cli>It also includes the First Derivative Test, comprising the concepts of local maxima, local minima, and point of inflexion.\u003c/li> \n \u003cli>It covers the Second Derivative Test and the working rule of finding absolute maxima and/or absolute minima.\u003c/li> \n\u003c/ul> \n\u003cp>To get access to the NCERT Solutions of all chapters of the Class 12 Maths with free PDF, and important topics, check - \u003ca href=\"https://www.shiksha.com/preparation/maths-ncert-solutions-class-12th-ns\">Class 12 Maths NCERT Solutions\u003c/a>.\u003c/p>","order":2,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"210580","status":"live","updateDateOn":false,"updatedOn":"2025-08-04 16:48:29","widgetData":[{"dataValue":"POLLS_WIDGET","order":1,"dataCategoryType":"widget","updateDateOn":false}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":902430,"h2Text":"Application of Derivatives NCERT Solution PDF: Free Download","dataValue":"\u003cp>Students should download the \u003cstrong>Application of Derivatives Class 12 PDF\u003c/strong> from the link given here. It will help them in getting concept clarity and scoring high in the \u003ca href=\"https://www.shiksha.com/boards/cbse-board\">CBSE Board\u003c/a> exam and in the \u003ca href=\"https://www.shiksha.com/engineering/jee-main-exam\">JEE Main exam\u003c/a>.\u003c/p> \n\u003cp>\u003cspan style=\"background-color: #fbeeb8;\">\u003cstrong>\u003cspan style=\"color: #e03e2d;\">Class 12 Math Chapter 6 Application of Derivatives Solution PDF\u003c/span>\u003c/strong>: \u003cstrong>\u003ca class=\"smce-docs smce-ftype-ico pdf \" style=\"background-color: #fbeeb8;\" href=\"https://images.shiksha.com/mediadata/pdf/1737377733phpxgKaJw.pdf\" rel=\"noopener\" target=\"_blank\">Free PDF Download\u003c/a> \u003c/strong>\u003c/span>\u003c/p>","order":3,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"180460","status":"live","updateDateOn":false,"updatedOn":"2025-05-16 18:06:24","widgetData":[{"id":950722,"dataValue":"PREP_ANA_FAQ","order":1,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"widget","wikiIdentifier":"210234","status":"live","updateDateOn":false,"updatedOn":"2025-08-01 12:59:04"}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":902428,"h2Text":"Class 12 Maths Application of Derivatives: Key Topics, Weightage ","dataValue":"\u003cp>\u003cstrong>Application of Derivatives Class 12\u003c/strong> is a part of calculus in JEE Main exams, which is a significant area of the Mathematics section. See below the topics covered in this chapter:\u003c/p> \n\u003ctable style=\"border-collapse: collapse; width: 68.3187%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003cth style=\"width: 22.0874%;\">Exercise\u003c/th> \n \u003cth style=\"width: 46.2322%;\">Topics Covered\u003c/th> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 22.0874%;\">6.1\u003c/td> \n \u003ctd style=\"width: 46.2322%;\">Introduction\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 22.0874%;\">6.2\u003c/td> \n \u003ctd style=\"width: 46.2322%;\">Rate of Change of Quantities\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 22.0874%;\">6.3\u003c/td> \n \u003ctd style=\"width: 46.2322%;\">Increasing and Decreasing Functions\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 22.0874%;\">6.4\u003c/td> \n \u003ctd style=\"width: 46.2322%;\">Maxima and Minima\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ch3>Application of Derivatives Class 12 Weightage in JEE Mains\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 65.0138%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003cth style=\"width: 23.1752%;\">Exam\u003c/th> \n \u003cth style=\"width: 38.0935%;\">Number of Questions\u003c/th> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 23.1752%;\">JEE Main\u003c/td> \n \u003ctd style=\"width: 38.0935%;\">2-4 questions\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ch3>Related Links\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 33.3333%;\">\u003ca href=\"https://www.shiksha.com/preparation/ncert-notes-tp\">NCERT Notes for Class 11 & 12\u003c/a>\u003c/td> \n \u003ctd style=\"width: 33.3333%;\">\u003ca href=\"https://www.shiksha.com/preparation/ncert-class-12-notes-tp\">NCERT Class 12 Notes\u003c/a>\u003c/td> \n \u003ctd style=\"width: 33.3333%;\">\u003ca href=\"https://www.shiksha.com/preparation/ncert-class-12-maths-tp\">Class 12 Maths Notes PDF for CBSE Exams\u003c/a>\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table>","order":4,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"180458","status":"live","updateDateOn":false,"updatedOn":"2025-05-16 18:06:24","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":952250,"h2Text":"Important Formulas of Application of Derivatives Class 12","dataValue":"\u003ch3>Application of Derivatives: Important Formulae for CBSE and Competitive Exams\u003c/h3> \n\u003cul> \n \u003cli>\u003cstrong>Rate of Change\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{dy}{dx}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>, where \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n y\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>is the dependent variable and \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cspan class=\"katex-html\" aria-hidden=\"true\">\u003cspan class=\"base\">\u003cspan class=\"mord mathnormal\">x\u003c/span>\u003c/span>\u003c/span>\u003c/span> is the independent variable.\u003c/li> \n \u003cli>\u003cstrong>Slope of Tangent\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n m = f'(x)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Slope of Normal\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n m = -\\frac{1}{f'(x)}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Equation of the tangent:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n y - y_1 = f'(x_1)(x - x_1)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Equation of the normal:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n y - y_1 = -\\frac{1}{f'(x_1)}(x - x_1)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Approximation Formula\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \approx\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f(x + h) \\approx f(x) + h \\cdot f'(x)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Condition for Increasing Function\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) > 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Condition for Decreasing Function\u003c/strong>: \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) \u003c 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Local Maxima/Minima (First Derivative Test)\u003c/strong>: Solve \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) = 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>and analyze sign changes of \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>around the critical points.\u003c/li> \n \u003cli>\u003cstrong>Conditions for Minima and Maxima:\u003c/strong> \n \u003cul> \n \u003cli style=\"list-style-type: none;\"> \n \u003cul> \n \u003cli>\u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) = 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>and \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f''(x) > 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>for local minima.\u003c/li> \n \u003cli>\u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) = 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>and \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f''(x) \u003c 0\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>for local maxima.\u003c/li> \n \u003c/ul> \u003c/li> \n \u003c/ul> \u003c/li> \n \u003cli>\u003cstrong>Second Derivative Test\u003c/strong>: Use \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003cmo mathvariant=\"normal\">\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f''(x)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>to confirm the nature of critical points.\u003c/li> \n\u003c/ul>","order":5,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"210582","status":"live","updateDateOn":false,"updatedOn":"2025-08-04 17:05:20","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":911042,"h2Text":"Class 12 Application of Derivatives Exercise 6.1 Solutions","dataValue":"\u003cp>Class 12 Application of Derivatives Exercise 6.1 discusses the fundamental usage of Derivative techniques to solve various kinds of problems. Application of Derivatives Exercise 6.1 focuses on the rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and finding maxima and minima. The concepts discussed in Class 12 Math Exercise 6.1 Solutions will help students get a good understanding of calculus and even provide tools to solve problems in Class 12 physics and other fields too. Class 12 Ch 6 Exercise 6.1 Solutions includes 18 Questions (10 Long, 6 Short, 2 MCQs). Students can check the complete solutions for Exercise 6.1 below;\u003c/p> \n\u003ch3>Application of Derivatives Exercise 6.1 Solutions\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q1. Find the rate of change of the area of a circle with respect to its radius r when\u003c/strong>\u003c/p> \u003cp>\u003cstrong>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> (π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>) = 2πr.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.1.\u003c/strong> (a) r = 3cm\u003c/p> \u003cp>When r = 3 cm,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 2 \times π \times 3 cm = 6π.\u003c/p> \u003cp>Thus, the area of the circle is changing at the rate of 6π cm \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>.\u003c/p> \u003cp>(b) r = 4cm\u003c/p> \u003cp>whenr = 4cm,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 2 \times π \times 4cm = 8π.\u003c/p> \u003cp>Thus, the area of the circle is changing at the rate of 8 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q2. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of an edge is 12 cm?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.2. \u003c/strong>Let x be the length of edge,v be the value and s be the surface area of the cube then,\u003c/p> \u003cp>y = x3.\u003c/p> \u003cp>and S = 6x2, where x is a fxn of time.\u003c/p> \u003cp>Now, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n Y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n F\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> (x3) = 8\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> (by chain rule)\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 3x2 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n S\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> (bx2) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 12x \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>.\u003c/p> \u003cp>When x = 12 cm,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n S\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.3. \u003c/strong>Let ‘r’ cm be the radius of the circle which is afxn of time.\u003c/p> \u003cp>Then, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 8 3cm/s as it is increasing.\u003c/p> \u003cp>Now, the area A of the circle is A = πr2.\u003c/p> \u003cp>So, the rate at which the area of the circle change \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> πr2.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 2πr 3\u003c/p> \u003cp>= 6πr. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n cm\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>When r = 10cm,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 6.π \times 10 = 60π \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n cm\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.4. \u003c/strong>Let ‘x’ cm be the length of edge of the cube which is a \u003cem>fx\u003csup>n\u003c/sup>\u003c/em> of time t then,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 3cm/s as it is increasing.\u003c/p> \u003cp>Now, volume v of the cube is v = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>\u003c/p> \u003cp>Ø Rate of change of volume of the cube \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n q\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.3\u003c/p> \u003cp>= 9\u003cem>x\u003c/em>\u003csup>2\u003c/sup> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n cm\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>When x = 10cm.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 9 x (10)\u003csup>2\u003c/sup>= 900 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n cm\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table>","order":6,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"180144","status":"live","updateDateOn":false,"updatedOn":"2025-06-03 15:27:30","widgetData":[{"id":911044,"dataValue":"ANA_FAQ","order":1,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"widget","wikiIdentifier":"198678","status":"live","updateDateOn":false,"updatedOn":"2025-06-03 15:27:36"}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":893002,"h2Text":"NCERT Solutions for Class 12 Maths Ch 6 Application of Derivatives- FAQs","dataValue":"\u003cp>Check few of the most asked questions related to this chapter by students below;\u003c/p>","order":7,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"194700","status":"live","updateDateOn":false,"updatedOn":"2025-04-30 13:25:40","widgetData":[{"id":893034,"dataValue":"ANA_FAQ","order":1,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"widget","wikiIdentifier":"194708","status":"live","updateDateOn":false,"updatedOn":"2025-04-30 13:50:01"}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":910806,"h2Text":"Class 12 Application of Derivatives 6.5 Solutions- (Old NCERT)","dataValue":"\u003cp>Class 12 Math Application of Derivatives Exercise 6.5 mainly focuses on problems such as finding minimum and maximum values of various types of functions, finding minima and maxima on a given point of functions, real real-world problems ( finding minimum and maximum profit), and finding values of volume, area, and more in a given interval for function. The Class 12 Exercise 6.5 Solutions will guide students about approaches, critical thinking skills, first and second derivative tests, and their applications in optimization. Application of Derivatives Exercise 6.5 consists of 29 Questions including 15 Long, 11 Short, and 3 MCQs. Students can check the complete solutions of the exercise 6.5 below;\u003c/p> \n\u003ch3>Application of Derivative Exercise 6.5 Solutions\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q74.Find the maximum and minimum values, if any, of the following functions\u003c/strong>\u003c/p> \u003cp>\u003cstrong>given by.\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A(i) \u003c/strong>We have, f(x) = (2\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3.\u003c/p> \u003cp>For all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>(2\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3 \geq 3.\u003c/p> \u003cp>f(x) \geq 3.\u003c/p> \u003cp>∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Again as \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values.\u003c/p> \u003cp>(ii) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(ii)\u003c/p> \u003cp>We have, f(x) = 9\u003csup>2\u003c/sup> + 12x + 2.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> (Taking 9 common from each team).\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>For all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x)\geq - 2.\u003c/p> \u003cp>∴The minimum value of f(x) = -2 when \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And as \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> so f(x) has \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>no maximum values.\u003c/p> \u003cp>(iii) f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 10\u003c/p> \u003cp>A(iii)\u003c/p> \u003cp>we have, f(x) = - (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 10\u003c/p> \u003cp>For all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> \leq 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> -(\u003cem>x\u003c/em>- 1)\u003csup>2\u003c/sup> + 10 \leq 10.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) \leq 10.\u003c/p> \u003cp>∴maximum value of f(x) = 10 when x - 1 = 0 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 1.\u003c/p> \u003cp>And minimum value of f(x) does not exist.\u003c/p> \u003cp>(iv) g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 1\u003c/p> \u003cp>A(iv)\u003c/p> \u003cp>we have, g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 1\u003c/p> \u003cp>For the given \u003cem>fx\u003c/em>\u003csup>n\u003c/sup>, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>and \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n \infty\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Q Maximum and minimum value does not exist.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q75. Find the maximum and minimum values, if any, of the following functions \u003c/strong>\u003cstrong>given by\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i) \u003c/strong>f(x) = |x + 2| - 1\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A(i) \u003c/strong>we have, f(x) = |x + 2| - 1\u003c/p> \u003cp>We know that, for all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x)\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>- 1.\u003c/p> \u003cp>∴ Minimum value of f(x) = -1 when x + 2 = 0 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = - 2.\u003c/p> \u003cp>And maximum value of f(x) does not exist.\u003c/p> \u003cp>(ii) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(ii)\u003c/p> \u003cp>We have, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>For all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n \geq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> g(x) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>3.\u003c/p> \u003cp>∴ Maximum value of g(x) = 3 when \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And minimum value does not exist.\u003c/p> \u003cp>(iii) h(x) = sin (2x) + 5.\u003c/p> \u003cp>A(iii)\u003c/p> \u003cp>we have, h(x) = sin (2x) + 5.\u003c/p> \u003cp>For all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> {range of sine function is [-1, 1]}\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> -1 + 5 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>sin 2x + 5 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>1 + 5.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 4 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>h(x) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>6.\u003c/p> \u003cp>∴ Maximum value of h(x) = 6.\u003c/p> \u003cp>Minimum value of h(x) = 4.\u003c/p> \u003cp>(iv) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(iv)\u003c/p> \u003cp>we have, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>As for all \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmi>\n ℝ\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> -1 + 3 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>sin 4x + 3\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 1 + 3\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n |\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 2 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>f(x) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>4.\u003c/p> \u003cp>∴ Maximum value of f(x) = 4.\u003c/p> \u003cp>Minimum value of f(x) = 2.\u003c/p> \u003cp>(v) h(x) = x + 1.,x\in ( -1, 1).\u003c/p> \u003cp>A(v)\u003c/p> \u003cp>we have, h(x) = x + 1.,x\in ( -1, 1).\u003c/p> \u003cp>Given, -1\n \u003cx\u003c1\u003c strong=\"\">\u003c/x\u003c1\u003c>\u003c/p> \u003cp>- 1 + 1 \u003c\u003cem>x\u003c/em> + 1 \u003c 1 + 1\u003c/p> \u003cp>0 \u003c\u003cem>h\u003c/em> (\u003cem>x\u003c/em>) \u003c 2.\u003c/p> \u003cp>∴ Maximum value and minimum value of h(x) does not exist.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%; height: 109px;\" border=\"1\"> \n \u003ctbody> \n \u003ctr style=\"height: 88px;\"> \n \u003ctd style=\"width: 100%; height: 88px; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q76. Find the local maxima and local minima, if any, of the following functions.\u003c/strong>\u003c/p> \u003cp>\u003cstrong> Find also the local maximum and the local minimum values, as the case may be:\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i) f(x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr style=\"height: 21px;\"> \n \u003ctd style=\"width: 100%; height: 21px;\"> \u003cp>\u003cstrong>A(i) \u003c/strong>we have, f(x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p> \u003cp>f(x) = 2x.\u003c/p> \u003cp>andf(x) = 2.\u003c/p> \u003cp>AR extreme point, f(x) = 0\u003c/p> \u003cp>2x = 0\u003c/p> \u003cp>x = 0.\u003c/p> \u003cp>When x = 0, f(0) = 2 > 0.\u003c/p> \u003cp>∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0\u003csup>2\u003c/sup> = 0.\u003c/p> \u003cp>(ii) g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> 3x\u003c/p> \u003cp>A(ii)\u003c/p> \u003cp>we have, g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 3\u003cem>x\u003c/em>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> g'(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> g''(x) = 6x.\u003c/p> \u003cp>At extreme point,\u003c/p> \u003cp>g'(x) = 0\u003c/p> \u003cp>3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3 = 0.\u003c/p> \u003cp>3(\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 1) = 0 \u003cspan style=\"background-color: #09818f; color: #ffffff; font-weight: 600;\">\Rightarrow \u003c/span>3(x - 1)(x + 1) = 0.\u003c/p> \u003cp>x = 1 or x = -1.\u003c/p> \u003cp>At x = 1, g\"(1) = 6.1 = 6 > 0.\u003c/p> \u003cp>So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1\u003csup>3\u003c/sup>- 3.1 = 1 - 3 = - 2.\u003c/p> \u003cp>And at x = -1, g\"( -1) = 6 ( -1) = 6 \u003c 0.\u003c/p> \u003cp>So, x = -1 is a point of local minima and value of local minimum is given by\u003c/p> \u003cp>g(- 1) = (- 1)\u003csup>3\u003c/sup>- 3(- 1) = 1 + 3 = 2.\u003c/p> \u003cp>(iii) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(iii)\u003c/p> \u003cp>we have, h(x) = sin x + cos x.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> h'(x) = cos x - sin x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> h'(x) = -sin x - cos x.\u003c/p> \u003cp>At extreme points,\u003c/p> \u003cp>h'(x) = 0\u003c/p> \u003cp>cosx - sin x = 0\u003c/p> \u003cp>cosx = sin x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 1 = tan x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"1200\" height=\"675\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 1200 675'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>(iv) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(iv)\u003c/p> \u003cp>we have, f(x) = sin x cos x.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = cos x + sin x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = sin x + cos x.\u003c/p> \u003cp>At extreme points,\u003c/p> \u003cp>f(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> cosx + sin x = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> IInd quadrate or \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> IIIth quadrate\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n A\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\u003c/mo>\n \u003cmo>\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n \u003cbr>\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\n \u003c/mo>\n \u003cmn>\u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"426\" height=\"371\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 426 371'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003cdiv class=\"figure\">\n (v) f(x) = \n \u003cem>x\u003c/em>\n \u003csup>3\u003c/sup>- 6\n \u003cem>x\u003c/em>\n \u003csup>2\u003c/sup> + 9\n \u003cem>x\u003c/em> + 15.\n \u003c/div> \n \u003c/div> \u003cp>A(v)\u003c/p> \u003cp>we have, f(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 9\u003cem>x\u003c/em> + 15.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 9.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f'(x) = 6x - 12.\u003c/p> \u003cp>At extreme point, f'(x) = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 9 = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 4\u003cem>x\u003c/em> + 3 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>-\u003cem>x \u003c/em>- 3\u003cem>x\u003c/em> + 3 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x(x - 1) -3(x - 1) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> (x - 1)(x - 3) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 1 or x = 3.\u003c/p> \u003cp>At, x = 1,f\"(1) = 6 \times 1 - 12 = 6 - 12 = - 6 \u003c 0\u003c/p> \u003cp>∴x = 1 is a point of local maxima and the value of local\u003c/p> \u003cp>Maximum is given by f(1) = 1\u003csup>3\u003c/sup>- 6(1)\u003csup>2\u003c/sup> + 9(1) + 15\u003c/p> \u003cp>= 1 - 6 + 9 + 15\u003c/p> \u003cp>= 19.\u003c/p> \u003cp>And at x = 3,f\"(3) = 6 \times 3 - 12 = 18 - 12 = 6 > 0\u003c/p> \u003cp>∴x = 3 is a point of local minima and the value of\u003c/p> \u003cp>local minimum is given by f (3) = 3\u003csup>3\u003c/sup>- 6(3)\u003csup>2\u003c/sup> + 9(3) + 15.\u003c/p> \u003cp>= 27 - 54 + 27 + 15.\u003c/p> \u003cp>= 15.\u003c/p> \u003cp>(vi) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(vi)\u003c/p> \u003cp>We have, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, extreme point, g'(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = -2 or x = 2\u003c/p> \u003cp>Given, that x > 0, hence we have x = 2.\u003c/p> \u003cp>At, x = 2, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmover accent=\"true\">\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo>\n ˙\n \u003c/mo>\n \u003c/mover>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>∴x = 2 is a point of local minima and value of local\u003c/p> \u003cp>Minimum is given by g(2) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>(vii) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A(vii)\u003c/p> \u003cp>Given, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At extreme points, g'(x) = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 0.\u003c/p> \u003cp>At, x = 0, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n '\n \u003c/mo>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>∴x = 0 is a point of local maxima and value of local maximum is given by \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"319\" height=\"295\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 319 295'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734069223phptw07x9_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"389\" height=\"462\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734069223phptw07x9.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 389 462'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q77. Prove that the following functions do not have maxima or minima:\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i) f(x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A(i) \u003c/strong>We have, f(x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n e\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, extreme points,\u003c/p> \u003cp>f(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>e\u003csup>x\u003c/sup>\u003c/em> = 0 which has no real ‘a’ value\u003c/p> \u003cp>∴f(x) has with maximum or minima\u003c/p> \u003cp>(ii) g(x) = log x\u003c/p> \u003cp>A(ii)\u003c/p> \u003cp>We have, g(x) = log x,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> g(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At extreme points,\u003c/p> \u003cp>g(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 1 = 0 which is not true.\u003c/p> \u003cp>∴g(x) was value minima or maxima\u003c/p> \u003cp>(iii) h(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>x\u003c/em> + 1.\u003c/p> \u003cp>A(iii)\u003c/p> \u003cp>We have, h(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>x\u003c/em> + 1.\u003c/p> \u003cp>h(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2x + 1\u003c/p> \u003cp>At extreme points,\u003c/p> \u003cp>h(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2x + 1 = 0\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734072775phpqEP6tT_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"211\" height=\"161\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734072775phpqEP6tT.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 211 161'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q78. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i) f(x) =\u003cem>x\u003c/em>\u003csup>3\u003c/sup> , x \in [- 2, 2]\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.(i) \u003c/strong>We have,\u003c/p> \u003cp>f(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> , x \in [- 2, 2].\u003c/p> \u003cp>f(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p> \u003cp>At, f(x) = 0\u003c/p> \u003cp>3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 0\u003c/p> \u003cp>x = 0 \u003c--[-2, 2].\u003c/p> \u003cp>We shall absolute the value of f at x = 0 and points of interval [ -2, 2]. So,\u003c/p> \u003cp>f(0) = 0\u003c/p> \u003cp>\u003cem>f\u003c/em>(- 2) = (- 2)\u003csup>3\u003c/sup> = 8\u003c/p> \u003cp>\u003cem>f\u003c/em>(2) = 2\u003csup>3\u003c/sup> = 8.\u003c/p> \u003cp>∴ Absolute maximum value of f(x) = 8 at x = 2\u003c/p> \u003cp>and absolute minimum value of f(x) = -8 at x = -2.\u003c/p> \u003cp>(ii) f (x) = sin x + cos x , x \in [0, π]\u003c/p> \u003cp>A.(ii)\u003c/p> \u003cp>We have, f(x) = sin x + cos x , x \in [0, π]\u003c/p> \u003cp>f(x) = cos x - sin x.\u003c/p> \u003cp>atf(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> cosx - sin x = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> sinx = cos x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n °\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734072978phpTrh5em_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"238\" height=\"109\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734072978phpTrh5em.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 238 109'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>(iii) f(x) = 4x \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>A.(iii)\u003c/p> \u003cp>We have, f(x) = 4x \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>f(x) = 4 - x\u003c/p> \u003cp>atf(x) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 4- x = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 4 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \in\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -2,\n \u003c/mo>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 7.87.5\u003c/p> \u003cp>Hence, absolute maximum value of f(x) = 8 at x = 4\u003c/p> \u003cp>and absolute minimum value of f(x) = -10 at x = -2.\u003c/p> \u003cp>(iv) f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3, \u003cem>x\in\u003c/em>[ -3, 1]\u003c/p> \u003cp>A.(iv)\u003c/p> \u003cp>We have, f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3, \u003cem>x\in\u003c/em>[ -3, 1]\u003c/p> \u003cp>f(x) = 2(x -1)\u003c/p> \u003cp>atf(x) = 0\u003c/p> \u003cp>2(x - 1) = 0\u003c/p> \u003cp>x = 1\u003cem>\in\u003c/em> [ -3, 1]\u003c/p> \u003cp>∴f(1) = (1 - 1)\u003csup>2\u003c/sup> + 3 = 3\u003c/p> \u003cp>\u003cem>f\u003c/em>(- 3) = (- 3 - 1)\u003csup>2\u003c/sup> + 3 = 16 + 3 = 1\u003c/p> \u003cp>∴ Absolute maximum value of f(x) = 19 at x = -3.\u003c/p> \u003cp>And absolute minimum value of f(x) = 3 at x = 1.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q79. Find the ma pro\u003c/strong>\u003cspan style=\"font-weight: 600;\">ximum\u003c/span>\u003cstrong>fit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x\u003csup>2\u003c/sup>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.6. \u003c/strong>We have, p(x) = 41 -\u003cem>f\u003c/em>2\u003cem>x \u003c/em>- 18\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p> \u003cp>P(x) = - 72 - 36\u003cem>x\u003c/em>\u003c/p> \u003cp>P(x) = -36\u003c/p> \u003cp>At extreme point,\u003c/p> \u003cp>- 72 - 36\u003cem>x\u003c/em> = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/p> \u003cp>At \u003cem>x\u003c/em> = - 2, \u003cem>p\"\u003c/em>(\u003cem>x\u003c/em>) = - 36 \u003c 0.\u003c/p> \u003cp>∴x = -2 is a point of local maximum and the value of local\u003c/p> \u003cp>Maximum is given by P(2) = 41 - 72(- 2) - 18(- 2)\u003csup>2\u003c/sup>\u003c/p> \u003cp>41 + 144 - 72 = 113 units.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q80. Find both the maximum value and the minimum value of 3x\u003csup>4\u003c/sup> - 8x\u003csup>3\u003c/sup> + 12x\u003csup>2\u003c/sup> - 48x + 25 on the interval [0, 3].\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A7. \u003c/strong>We have, f(x) = 3\u003cem>x\u003c/em>\u003csup>4\u003c/sup>- 8\u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 12\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 48\u003cem>x\u003c/em> + 25, \u003cem>x \in\u003c/em> [0, 3].\u003c/p> \u003cp>f(x) = 12\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 24\u003cem>x \u003c/em>- 48.\u003c/p> \u003cp>At f(x) = 0.\u003c/p> \u003cp>12\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 24\u003cem>x \u003c/em>- 48 = 0.\u003c/p> \u003cp>\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 2\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2\u003cem>x \u003c/em>- 4 = 0\u003c/p> \u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup>(\u003cem>x \u003c/em>- 2) + 2(\u003cem>x \u003c/em>- 2) = 0\u003c/p> \u003cp>(\u003cem>x \u003c/em>- 2) + (\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2) = 0\u003c/p> \u003cp>x = 2 \in [0, 3] or x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \pm\sqrt-2\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> which is not possible as\u003c/p> \u003cp>∴f(x) = 3(2)\u003csup>4\u003c/sup>- 8(2)\u003csup>3\u003c/sup> + 12(2)\u003csup>2\u003c/sup>- 4(2) + 25.\u003c/p> \u003cp>=48 - 64 + 48 - 96 + 25.\u003c/p> \u003cp>= -39.\u003c/p> \u003cp>f(0) =3(0)\u003csup>4\u003c/sup>- 8(0)\u003csup>3\u003c/sup> + 12(0)\u003csup>2\u003c/sup>- 48(0) + 25.\u003c/p> \u003cp>= 25.\u003c/p> \u003cp>f(3) = 3(3)\u003csup>4\u003c/sup>- 8(3)\u003csup>3\u003c/sup> + 12(3)\u003csup>2\u003c/sup>- 48(3) + 25.\u003c/p> \u003cp>= 243 - 216 + 108 - 144 + 25\u003c/p> \u003cp>= 16.\u003c/p> \u003cp>Maximum value of f(x) = 25 at x = 0.\u003c/p> \u003cp>and minimum value of f(x) = -39 at x = 2.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q81. At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A8. \u003c/strong>We have, f(x) = sin 2x, x \in [0, 2π],\u003c/p> \u003cp>f(x) = 2cos 2x.\u003c/p> \u003cp>At f(x) = 0.\u003c/p> \u003cp>2 cos 2x = 0\u003c/p> \u003cp>cos 2x = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 1. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 1.\u003c/p> \u003cp>f(0) = sin 2(0) = sin 0 = 0\u003c/p> \u003cp>f(2π) = sin 2(2π) = sin 4π = 0\u003c/p> \u003cp>Hence, the points of maximum \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmi>\n x are.\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )and \n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q82. What is the maximum value of the function sin x + cos x?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A9. \u003c/strong>We have, f(x) = sin x + cos x.\u003c/p> \u003cp>f(x) = cos x - sin x.\u003c/p> \u003cp>At f(x) = 0\u003c/p> \u003cp>cosx - sin x = 0\u003c/p> \u003cp>sinx = cos x\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> ,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734073563phpXVSB7z_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"262\" height=\"107\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734073563phpXVSB7z.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 262 107'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cstrong>\u003cspan style=\"font-size: 0.875rem;\">Q83. Find the maximum value of 2x\u003c/span>\u003csup>3\u003c/sup>\u003cspan style=\"font-size: 0.875rem;\"> - 24x + 107 in the interval [1, 3].\u003c/span>\u003c/strong>\n \u003c/div> \n \u003c/div> \u003cp>\u003cstrong>Find the maximum value of the same function in [-3, -1]\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.10. \u003c/strong>We have, f(x) =2\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em> + 107, \u003cem>x\u003c/em>[1,3]\u003c/p> \u003cp>f(x) = 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 24.\u003c/p> \u003cp>At f(x) = 0\u003c/p> \u003cp>6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 24= 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = \pm2. -->x = 2 \in[1, 3].\u003c/p> \u003cp>So, f(2) = 2(2)\u003csup>3\u003c/sup>- 24(2) + 107 = 16 - 48 + 107 = 75.\u003c/p> \u003cp>f(1) = 2(1)\u003csup>3\u003c/sup>- 24(1) + 107 = 2 - 24 + 107 = 85.\u003c/p> \u003cp>f(3) = 2(3)\u003csup>3\u003c/sup>- 24(3) + 107 = 54 - 72 + 107 = 89\u003c/p> \u003cp>∴ Maximum value of f(x) in interval [1, 3] is 89 at x = 3.\u003c/p> \u003cp>When x \in [ -3, -1]\u003c/p> \u003cp>From f(x) = 0\u003c/p> \u003cp>x = -2 \in [ -3, -1]\u003c/p> \u003cp>So, \u003cem>f\u003c/em>(- 2) = 2(- 2)\u003csup>3\u003c/sup>- 24(- 2) + 107 = - 16 + 48 + 107 = 139.\u003c/p> \u003cp>\u003cem>f\u003c/em>(- 3) = 2(- 3)\u003csup>3\u003c/sup>- 24(- 3) + 107 = - 54 + 72 + 107 = 125.\u003c/p> \u003cp>\u003cem>f\u003c/em>(- 1) = 2(- 1)\u003csup>3\u003c/sup>- 24(- 1) + 107 = - 2 + 24 + 107 = 129.\u003c/p> \u003cp>∴ Maximum value of f(x) in interval [ -3, -1] is 139 at x = -2.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cdiv class=\"photo-widget-full\">\n \n\u003c/div> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q84. It is given that at x = 1, the function x\u003csup>4\u003c/sup> - 62x\u003csup>2\u003c/sup> + ax + 9 attains \u003c/strong>\u003cstrong>its maximum value, on the interval [0, 2]. Find the value of a.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.11. \u003c/strong>We have, f(x) = \u003cem>x\u003c/em>\u003csup>4\u003c/sup>- 62\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>ax\u003c/em> + 9, \u003cem>x\in\u003c/em> [0, 2].\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 124\u003cem>x\u003c/em> + \u003cem>a\u003c/em>\u003c/p> \u003cp>∴f(x) active its max\u003csup>n\u003c/sup> value at x = 1\u003cem>\in\u003c/em> [0, 2]\u003c/p> \u003cp>∴f(1) = 0.\u003c/p> \u003cp>4(1)\u003csup>3\u003c/sup>- 124(1) + a = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> a = 124 - 4 = 120.\u003c/p> \u003cp>∴a = 120\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q85. Find the maximum and minimum values of x + sin 2x on [0, 2π].\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.12. \u003c/strong>We have, f(x) = x + sin 2x ,x \in [0, 2π].\u003c/p> \u003cp>f(x) = 1 + 2cos 2x\u003c/p> \u003cp>At f(x) = 0\u003c/p> \u003cp>1 + 2 cos2x = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Missing\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n +\sqrt3/2\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 1.05 + 0.87\u003c/p> \u003cp>= 1.92\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 1.23\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>=5.07.\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 5.25 - 0.87 = 4.38\u003c/p> \u003cp>At and points,\u003c/p> \u003cp>f(0) = 0 + sin2 \times 0 = 0\u003c/p> \u003cp>f(2π) = 2π + sin 2 \times 2π = 6.2 + 0 = 6.28\u003c/p> \u003cp>∴Maximum value of f(x) = 6.28 at x = 2π and\u003c/p> \u003cp>minimum value of f(x) = 0 at x= 0\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q86. Find two numbers whose sum is 24 and whose product is as large as possible.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.13. \u003c/strong>Let ‘x’ and ‘y’ be the two number\u003c/p> \u003cp>Then, x + y = 24 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> y = 24 - x\u003c/p> \u003cp>Let ‘P’ be their product then,\u003c/p> \u003cp>P = xy = x (24 - x) = 24\u003cem>x \u003c/em>-\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>P(x) = 24\u003cem>x \u003c/em>-\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>24 - 2x = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, P(12) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = 12 is a point of local maxima\u003c/p> \u003cp>Hence, y = 24 - 12 = 12.\u003c/p> \u003cp>The uqdtwno (x, y) is (12, 12).\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q87. Find two positive numbers x and y such that x + y = 60 and xy\u003csup>3\u003c/sup> is maximum.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.14. \u003c/strong>We have, x + y = 60.where x, y > 0\u003c/p> \u003cp>x = 60 - y.\u003c/p> \u003cp>Let the product P = \u003cem>xy\u003c/em>\u003csup>3\u003c/sup> = (60 -\u003cem>y\u003c/em>) \u003cem>y\u003c/em>\u003csup>3\u003c/sup> = 60\u003cem>y\u003c/em>\u003csup>3\u003c/sup>-\u003cem>y\u003c/em>\u003csup>4\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= 4\u003cem>y\u003c/em>\u003csup>2\u003c/sup>(45 -\u003cem>y\u003c/em>)\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 4\u003cem>y\u003c/em>\u003csup>2\u003c/sup>(45 -\u003cem>y\u003c/em>) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> y = 0 and y = 45\u003c/p> \u003cp>As y > 0, y = 45\u003c/p> \u003cp>When, y > 45, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= ve \u003c 0\u003c/p> \u003cp>Ad y \u003c 45, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= (+ ve) > 0\u003c/p> \u003cp>∴p is maximum when y = 45 from + ve to- ve or y increases through 45.\u003c/p> \u003cp>So, x = 60 - y = 6Ø - 45 = 15.\u003c/p> \u003cp>Øx = 15 and y = 45.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q88. Find two positive numbers x and y such that their sum is 35 and the product \u003c/strong>\u003cstrong>x\u003csup>2 \u003c/sup>y\u003csup>5 \u003c/sup> is a maximum.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.15. \u003c/strong>We have, x + y = 35.\u003c/p> \u003cp>y = 35 - x\u003c/p> \u003cp>Let the product, P =x\u003csup>2 \u003c/sup>y\u003csup>5 \u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> P = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> (35 -\u003cem>x\u003c/em>)\u003csup>5\u003c/sup>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> 5(35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (1) + (35 -\u003cem>x\u003c/em>)\u003csup>5\u003c/sup> 2\u003cem>x\u003c/em>\u003c/p> \u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup>[ - 5\u003cem>x\u003c/em> + (35 -\u003cem>x\u003c/em>) 2]\u003c/p> \u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> [ - 5\u003cem>x\u003c/em> + 70 - 2\u003cem>x\u003c/em>]\u003c/p> \u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (70 - 7\u003cem>x\u003c/em>)\u003c/p> \u003cp>= 7\u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (10 -\u003cem>x\u003c/em>)\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>7\u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (10 -\u003cem>x\u003c/em>) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 0, 35, 10\u003c/p> \u003cp>As x is a (+) ve number we have only\u003c/p> \u003cp>x = 10, 35\u003c/p> \u003cp>And again (at x = 35) y = 35 = 0 but yis also a (+) ve number\u003c/p> \u003cp>we get, x = 10 (only)\u003c/p> \u003cp>whenx \u003c 10,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∵\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>and when x > 10,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> changes from (+ ve) to ( -ve) as x increases while passing through 10\u003c/p> \u003cp>Hence, x = 10 is a point of local maxima\u003c/p> \u003cp>So, y = 35 - 10 = 25\u003c/p> \u003cp>∴x = 10 and y = 25\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q89. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.16. \u003c/strong>Let, x and y be the two positive number\u003c/p> \u003cp>Then, x + y = 16 y = 16 - x\u003c/p> \u003cp>Let p be the sun of the cubes then\u003c/p> \u003cp>p = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>y\u003c/em>\u003csup>3\u003c/sup> = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + (16 -\u003cem>x\u003c/em>)\u003csup>3\u003c/sup> = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + (16)\u003csup>3\u003c/sup>-\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 48\u003cem>x\u003c/em> (16 -\u003cem>x\u003c/em>)\u003c/p> \u003cp>p = 16\u003csup>3\u003c/sup> + 48\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 76 8\u003cem>x\u003c/em>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>96x - 768 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>∴at x = 8, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n p\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, x = 8 is a point of local minima\u003c/p> \u003cp>So, y = 16 - 8 = 8\u003c/p> \u003cp>Hence, x = 8, y = 8\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q90. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.17. \u003c/strong>Side of the tin square piece = 18 cm\u003c/p> \u003cp>Let x cm be the thought of the square to be cut from each corner.\u003c/p> \u003cp>The volume v of the box after cutting\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> v = length \times breadth \times height\u003c/p> \u003cp>= (18 - 2\u003cem>x\u003c/em>)(18 - 2\u003cem>x\u003c/em>) \u003cem>x \u003c/em>\u003cem>x\u003c/em>\u003c/p> \u003cp>= (18 - 2\u003cem>x\u003c/em>)\u003csup>2\u003c/sup>\u003cem>x\u003c/em>\u003c/p> \u003cp>= (324 + 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 72\u003cem>x\u003c/em>) \u003cem>x\u003c/em>\u003c/p> \u003cp>= 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 72\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 324\u003cem>x\u003c/em>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>As \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 27 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 9\u003cem>x\u003c/em>- 3\u003cem>x\u003c/em> + 27 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em> (\u003cem>x \u003c/em>- 9) - 3 (\u003cem>x \u003c/em>- 9) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> (\u003cem>x \u003c/em>- 9)(\u003cem>x \u003c/em>- 3) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 9 and x = 3\u003c/p> \u003cp>At x = 0, length of box = 18 - 2π9 = 18 - 18 = 0\u003c/p> \u003cp>Which is not possible\u003c/p> \u003cp>And at x = 3, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 24(3) - 144 = -72 \u003c 0\u003c/p> \u003cp>∴x = 3 is a point of maximum\u003c/p> \u003cp>Hence, ‘3’ cm (square) is to be cut from each side of the square\u003c/p> \u003cp>So that volume of box is maximum\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q91. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.18. \u003c/strong>Let ‘x’ cm be the length of side of the square to be cut off from the rectangular surface\u003c/p> \u003cp>Then, the volume v of the box is v = (45 - 2\u003cem>x\u003c/em>)(24 - 2\u003cem>x\u003c/em>) \u003cem>x\u003c/em>\u003c/p> \u003cp>= 1080\u003cem>x \u003c/em>- 138\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \to\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 23\u003cem>x\u003c/em> + 90 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 5\u003cem>x \u003c/em>- 18\u003cem>x\u003c/em> + 90 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>(\u003cem>x \u003c/em>- 5) - 18 (\u003cem>x \u003c/em>- 5) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> (\u003cem>x\u003c/em>- 5) (\u003cem>x\u003c/em>- 18) = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> x = 5 and x = 18\u003c/p> \u003cp>At x = 18, breadth = 24 - 2(18) = 24 - 36 = -12 which is not possible\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, x = 5 is the point of maximum\u003c/p> \u003cp>So, ‘5’ cm length of square seeds to be cut from each corner of the secthgle\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q92. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.19. \u003c/strong>Let A.B.C.D.be the square increased in a given fixed circle with radius x\u003c/p> \u003cp>Let ‘x’ and ‘y’ be the length and breadth of the rectangle\u003c/p> \u003cp>∴x, y> 0\u003c/p> \u003cp>In ABC, right angle at B,\u003c/p> \u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = (2\u003cem>x\u003c/em>)\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082028phpxXg4rZ_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"283\" height=\"254\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082028phpxXg4rZ.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 283 254'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082043php6kapSo_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"362\" height=\"122\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082043php6kapSo.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 362 122'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q93. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.20. \u003c/strong>Let r and h be the radius and height of the cylinder\u003c/p> \u003cp>So, r, h > 0\u003c/p> \u003cp>The total surface area s is given by\u003c/p> \u003cp>S = 2πr(h + r) = content .\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n S\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> = content = x (say)\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n u\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Then, the volume v of the cylinder\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n V\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>For maximum, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082150php0lrnv8_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"246\" height=\"266\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082150php0lrnv8.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 246 266'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q94. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A21. \u003c/strong>The volume v of a cylinder of height h and radius r is\u003c/p> \u003cp>V = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003cem>h\u003c/em> = 100\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Let, s be the surface area then\u003c/p> \u003cp>S = 2\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003cem>h\u003c/em>r(r + h) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n S\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n S\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \RightarrowM\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n M\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2S\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> isa point of minimum\u003c/p> \u003cp>And \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q95. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.22. \u003c/strong>Let x and y in ‘m’ be the length of side of the square the radius of the circle respectily\u003c/p> \u003cp>Then, length of wire = perimeter of square + circumference of circle\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 28 = 4x + 2πy\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> 2x + πy = 14\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The combine area A of the square and the circle is\u003c/p> \u003cp>A = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + π\u003cem>y\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n dA\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2A\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n dA\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2A\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n γ\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> isa point of minima\u003c/p> \u003cp>Hence, length of square = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>and length of circle = 2πy\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q96. Prove that the volume of the largest cone that can be inscribed in a sphere \u003c/strong>\u003cstrong>of radius R is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> of the volume of the sphere.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.23. \u003c/strong>Let r and h be the radius and height of the one in scribed in the sphere of radius R.\u003c/p> \u003cp>Then, is ΔOBC, rt angle at B (\u003cem>h\u003c/em>-\u003cem>r\u003c/em>)\u003csup>2\u003c/sup> + \u003cem>r\u003c/em>\u003csup>2\u003c/sup> = R\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>h\u003c/em>\u003csup>2\u003c/sup> + R\u003csup>2\u003c/sup>- 2\u003cem>h\u003c/em>R + \u003cem>h\u003c/em>\u003csup>2\u003c/sup> = R\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cem>r\u003c/em>\u003csup>2\u003c/sup> = 2\u003cem>h\u003c/em>R -\u003cem>h\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>Then the volume v of the cone is, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n V\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmi>\n R\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n Rh\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n dV\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n Rh\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2V\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n dV\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>4Rh - 3\u003cem>h\u003c/em>\u003csup>2\u003c/sup> = 0.\u003c/p> \u003cp>h(4R - 3h) = 0.\u003c/p> \u003cp>h = 0 and \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n R\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>As h> 0, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2v\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n R\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4rR\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n R]\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n Rπ\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> is a point of maxima.\u003c/p> \u003cp>and \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n hR\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n R-\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 8R\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6R\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8R\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, Volume of Cone, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8R\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4R\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n πR\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> Volume of sphere.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q97. Show that the right circular cone of least curved surface and given volume \u003c/strong>\u003cstrong>has an altitude equal to \sqrt2\u003c/strong>\n \u003cmath>\n \u003cmrow>\u003c/mrow>\n \u003c/math>\u003cstrong> time the radius of the base.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.24. \u003c/strong>Let r and h be the radius and height of the cone.\u003c/p> \u003cp>The volume V of the cone is.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3V\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n k(SAY)\n \u003c/mi>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And curve surface area S is\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082893php6RewPt_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"523\" height=\"634\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734082893php6RewPt.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 523 634'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q98. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> \sqrt2.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.25. \u003c/strong>Let r, h, l and Ø be the radius, height, slant height and semi-vertical angle respectively of the cone. i.e., r, h, l>0.\u003c/p> \u003cp>Then, Volume V of the cone is\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n \u003cbr>\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n dV\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2V\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn> \n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p>\n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083051phpEK0yM8_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"348\" height=\"430\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083051phpEK0yM8.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 348 430'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp> \u003c/p> .\u003cp>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q99. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.26.\u003c/strong>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083769phptJNfSs_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"574\" height=\"265\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083769phptJNfSs.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 574 265'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083797php9uWQPa_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"505\" height=\"649\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083797php9uWQPa.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 505 649'>\u003c/svg>\">\n \u003c/picture> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083815phpLheqCo_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"244\" height=\"408\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083815phpLheqCo.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 244 408'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\"> \n \u003cp>\u003cstrong>Choose the correct answer in Questions 27 and 29.\u003c/strong>\u003c/p> \n \u003c/div> \n \u003c/div> \u003cp> \u003c/p> \u003cp>\u003cstrong>Q100. The point on the curve x2 = 2y which is nearest to the point (0, 5) is\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(A)\u003c/strong> (2\sqrt2 \n \u003cmath>\n \u003cmrow>\u003c/mrow>\n \u003c/math> ,4) \u003cstrong>(B)\u003c/strong> (2\sqrt2 \n \u003cmath>\n \u003cmrow>\u003c/mrow>\n \u003c/math> ,0) \u003cstrong>(C)\u003c/strong> (0, 0) \u003cstrong>(D)\u003c/strong> (2, 2)\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.27. \u003c/strong>The equation of the given curve is\u003c/p> \u003cp>\u003cem>x\u003csup>2\u003c/sup>\u003c/em> = 2y.\u003c/p> \u003cp>Let p(x, y) be a point on the curve.\u003c/p> \u003cp>The distance of p(x, y) from (0, 5) is say S is given by\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083902phphfzVA2_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"1200\" height=\"675\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734083902phphfzVA2.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 1200 675'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp> \u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Let z = \u003cem>s\u003c/em>\u003csup>2\u003c/sup> = \u003cem>x\u003c/em>\u003csup>2 \u003c/sup>+ \u003cem>y\u003c/em>\u003csup>2\u003c/sup> + 25 - 10\u003cem>y\u003c/em> = 2\u003cem>y\u003c/em> + \u003cem>y\u003c/em>\u003csup>2 \u003c/sup>-10\u003cem>y\u003c/em> + 25\u003c/p> \u003cp>z = \u003cem>y\u003c/em>\u003csup>2\u003c/sup> - 8\u003cem>y\u003c/em> + 25\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At y = 4, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>y = 4, is point of minimum distance.\u003c/p> \u003cp>So, \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 2\u003cem>y\u003c/em>-->\u003cem>x\u003c/em>2 = 2 \times 4--> \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 8 \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the point of nearest distance are \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\sqrt2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> and\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\sqrt2\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Option (A) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \n \u003c/div> \n\u003c/div> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q101. For all real values of x, the minimum value of \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> is\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(A)\u003c/strong> 0 \u003cstrong>(B)\u003c/strong> 1 \u003cstrong>(C)\u003c/strong> 3 \u003cstrong>(D)\u003c/strong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.28. \u003c/strong>We have,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmenclose notation=\"updiagonalstrike\">\n \u003cmi>\n f\n \u003c/mi>\n \u003c/menclose>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At f(x) = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = 1 and x = -1.\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At x = -1, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The maximum value of f(x) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, option (D) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%; height: 834px;\" border=\"1\"> \n \u003ctbody> \n \u003ctr style=\"height: 119px;\"> \n \u003ctd style=\"width: 100%; background-color: #ced4d9; height: 119px;\"> \u003cp>\u003cstrong>Q102. The maximum value of \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> , 0\n \u003cx\u003c1 is\u003c=\"\" strong=\"\">\u003c/x\u003c1>\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(A)\u003c/strong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cstrong>(B)\u003c/strong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cstrong>(C)\u003c/strong> 1 \u003cstrong>(D)\u003c/strong> 0\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr style=\"height: 715px;\"> \n \u003ctd style=\"width: 100%; height: 715px;\"> \u003cp>\u003cstrong>A.29. \u003c/strong>We have,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmi>\n X(x - 1)\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n +\n \u003c/mi>\n \u003cmo>\n \u003cbr>\n \u003c/mo>\n \u003cmn>\n 1]\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At f(x) = 0.\u003c/p> \u003cp>2x - 1 = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n ≃\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 0(0 - 1)\n \u003c/mn>\n \u003cmo>\n \u003cbr>\n \u003c/mo>\n \u003cmn>\n +\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n (1 - 1)\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1]\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n \u003cbr>\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Option (B) is\u003c/p> \u003cp>Hence maximum value of f(x) = at x = 1 and x = 0.\u003c/p> \u003cp>Option (c) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cp> \u003c/p>","order":8,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"109015","status":"live","updateDateOn":false,"updatedOn":"2025-06-03 12:41:20","h2CategoryId":29,"h2CategoryText":"faqs","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":910826,"h2Text":"NCERT Class 12 Application of Derivatives Miscellaneous Exercise Solutions","dataValue":"\u003cp>The Class 12 Application of Derivatives Miscellaneous Exercise contains problems based on a range of concepts covered in the chapter, including the rate of change, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, and optimization problems. This exercise is designed to test students' understanding and ability to apply these concepts in a variety of challenging situations. Miscellaneous Exercise Solutions will have answers to 24 Questions including 14 Long, 4 Short, and 6 MCQs. Students can check the complete solutions of the Chapter 6 Miscellaneous Exercise below;\u003c/p> \n\u003ch3>Application of Derivatives Miscellaneous Exercise Solution\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q103. Using differentials, find the approximate value of each of the following\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(a) \u003c/strong>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.1.\u003c/strong> (a) consider \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmi>\n L\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n &\n \u003c/mo>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n T\n \u003c/mi>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> is approximately equal to \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> and is given by,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the approximate value of \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>=0.677\u003c/p> \u003cp>(b) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>(b) Consider \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmi>\n L\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n &\n \u003c/mo>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087182phpQyBEuv_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"541\" height=\"148\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087182phpQyBEuv.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 541 148'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>Now, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> is approximately equal to \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> and is given by,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n △\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the approximate value of \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n 33\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n \u003cbr>\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q104. Show that the function given by \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> has maximum at x = e.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A2. \u003c/strong>We have, f(x) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>f(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmunder accentunder=\"true\">\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo stretchy=\"true\">\n _\n \u003c/mo>\n \u003c/munder>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>f(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At extreme points, f(x) = 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At x = e, f\"(e) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n e\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n e\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n e\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = e is a point of maximum.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q105. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A3. \u003c/strong>Let ‘b’ and ‘x’ be the fixed base and equal side of isosceales triangle.\u003c/p> \u003cp>Then, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> cm/s (Ø decreasing).\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087295php6A19As_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"385\" height=\"479\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087295php6A19As.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 385 479'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q106. Find the equation of the normal to curve x\u003csup>2 \u003c/sup>= 4y which passes through the point (1, 2)\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A4. \u003c/strong>Equation of the curve is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, the slope of the normal at point \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> Equation of the normal at \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q107. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ - aθ cosθ is at a constant distance from the origin.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A5. \u003c/strong>We have \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> Slope of the normal at any point \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The equation of the normal at a given point \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> is given by,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, the perpendicular distance of the normal from the origin is\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087410phpZY1Fs7_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"396\" height=\"70\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734087410phpZY1Fs7.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 396 70'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \n \u003cdiv>\n \u003cstrong>\u003cspan style=\"font-size: 0.875rem;\">Q108. Find the intervals in which the function f given by\u003c/span>\u003c/strong>\n \u003c/div> \n \u003cdiv> \n \u003cp>\u003cstrong>f(x)= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> is (i) increasing (ii) decreasing\u003c/strong>\u003c/p> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A6. \u003c/strong>We have, f(x)= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n s\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> as cos x lies in [1, 1].\u003c/p> \u003cp>So, (i) for increasing, f(x) \geq 0.\u003c/p> \u003cp>cosx \geq 0.\u003c/p> \u003cp>x lies in Ist and IVth quadrant.\u003c/p> \u003cp>i.e., f(x) is increasing for \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> and \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>(ii) for decreasing, f(x) \leq 0.\u003c/p> \u003cp>cosx \leq 0.\u003c/p> \u003cp>x lies in IInd and IIIrd quadrant.\u003c/p> \u003cp>i.e., f(x) is decreasing for \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \leq\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cdiv class=\"photo-widget-full\"> \n \u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q109. Find the intervals in which the function f given by \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \neq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> is\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(i)\u003c/strong> Increasing \u003cstrong>(ii)\u003c/strong> Decreasing\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A7. \u003c/strong>We have, f(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \neq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> { \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n b\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n {\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n ∵\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n b\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n b\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n \n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n \neq\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So we have three disjoint internal i.e.,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n \infty\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n \infty\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>When, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n \infty\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo>\n \n \u003c/mo>\n \u003cmtext>\n and\n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n \n \u003c/mo>\n \u003cmtext>\n at\n \u003c/mtext>\n \u003cmo>\n \n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, f(x) is increasing.\u003c/p> \u003cp>When \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n '\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n \n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n at\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1and \n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, f(x) is decreasing.\u003c/p> \u003cp>When \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \in\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n \infty\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>f(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmtext>\n on\n \u003c/mtext>\n \u003cmo>\n \n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n at\n \u003c/mo>\n \u003cmo>\n \n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, f(x) is increasing.\u003c/p> \u003cp>f(x) is increasing for x\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \in\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>(\infty,1) and [1, \infty] and decreasing for x\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \in\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>[1, 1].\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n \u003c/table> \n \u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q110. Find the maximum area of an isosceles triangle inscribed in the ellipse\u003c/strong>\u003c/p> \u003cp>\u003cstrong>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003c/mrow>\n \u003c/math>\u003c/span> with its vertex at one end of the major axis.\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.8. \u003c/strong>The given equation of the ellipse is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003c/mrow>\n \u003c/math>\u003c/span> (1)\u003c/p> \u003cp>Let the major axis be along x-axis so, vertex is at \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Let ΔABC be the isosceles triangle inscribed on the\u003c/p> \u003cp>ellipse with one vertex C at (a, 0).\u003c/p> \u003cp>Then, let A have Co-ordinate (x0, yo) from figure.\u003c/p> \u003cp>So, Co-ordinate of B = (x0, y0)\u003c/p> \u003cp>As A and B lies on the ellipse, from equation (i),\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088430phpEdYX3B_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"377\" height=\"290\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088430phpEdYX3B.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 377 290'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088443phpfqyN4H_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"398\" height=\"473\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088443phpfqyN4H.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 398 473'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088456phpW5DTG6_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"296\" height=\"386\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088456phpW5DTG6.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 296 386'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088469phpyQ2ZWZ_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"231\" height=\"284\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088469phpyQ2ZWZ.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 231 284'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n \u003c/table> \n \u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q111. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m\u003csup>3\u003c/sup>. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.9. \u003c/strong>Let x and y meters be the length and breath of the rectangular base of the tank respectively.\u003c/p> \u003cp>Then, volume V of the tank is\u003c/p> \u003cp>V = length \times depth \times breath.\u003c/p> \u003cp>V = 2xy = 8m\u003csup>3\u003c/sup>(given).\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Let ‘t’ be the total cost of building the tank.\u003c/p> \u003cp>Then, t = cost of base + cost of sides.\u003c/p> \u003cp>= 70xy + 45[4x+4y] {there are four sides.\u003c/p> \u003cp>= 70xy + 180x+ 180y.\u003c/p> \u003cp>= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n z\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = \pm 2\u003c/p> \u003cp>x = 2, (x = length and it cannot be negative)\u003c/p> \u003cp>At x = 2, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = 2 is point of maxima.\u003c/p> \u003cp>Hence, minimum cost = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 280 + 360 + 360 = 1000.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n \u003c/table> \n \u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q112. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.10. \u003c/strong>Let r and s be the radius of the circle and length of side of square.\u003c/p> \u003cp>Then, sum of perimeter of circle and square = k\u003c/p> \u003cp>2πr +4s = k\u003c/p> \u003cp>s = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The area A be the total areas of the circle and square.\u003c/p> \u003cp>Then, A = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup> + \u003cem>s\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n A\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2A\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n A\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2A\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088717php8O6Esh_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"239\" height=\"158\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734088717php8O6Esh.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 239 158'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>s = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>s = 2r.\u003c/p> \u003cp>Hence, proved.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n \u003c/table> \n \u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q113. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.11. \u003c/strong>Let ‘x’ metre be the radius of the semi-circular opening mounded on the length ‘2x’ side of rectangle. Then, let ‘y’ be the breadth of the rectangle.\u003c/p> \u003cp>Then, perimeter of the window = 10m\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x + 2x + 2y + = 10.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Let the area of the window be A.\u003c/p> \u003cp>Then, A = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> [-π\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 20\u003cem>x\u003c/em>].\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> [ -2πx - 8x + 20]\u003c/p> \u003cp>And \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmover>\n \u003cmrow>\u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mover>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> [ -2π - 8] = -π -4 = -( π+ 4)\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> [ -2πx - 8x + 20] = 0\u003c/p> \u003cp>2x + 8x = 20\u003c/p> \u003cp>x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2A\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = -( π+ 4) \u003c 0\u003c/p> \u003cp>Øx = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> is a point of minima.\u003c/p> \u003cp>And y = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Ø Dimensions of the window are\u003c/p> \u003cp>length = 2x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>breadth = y \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>radius = y = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n \u003c/table> \n\u003c/div> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q114. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is\u003c/strong>\u003c/p> \u003cp>\u003cstrong>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.12. \u003c/strong>Let P be the point on hypotenuse of a triangle. ABC, t angle at B.\u003c/p> \u003cp>Which is at distance a& b from the sides of the triangle.\u003c/p> \u003cp>Let \u003c BAC = \u003c MPC = Ø.\u003c/p> \u003cp>Then, in\dots ΔANP,\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089111phpN0pSvT_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"324\" height=\"449\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089111phpN0pSvT.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 324 449'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmi>\n θ\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At tan Ø = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmover accent=\"true\">\n \u003cmi>\n q\n \u003c/mi>\n \u003cmo>\n ˙\n \u003c/mo>\n \u003c/mover>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> { Øall trigonometric f\u003cem>x\u003c/em>\u003csup>n\u003c/sup> are + ve in I\u003csup>st\u003c/sup> quadrant}.\u003c/p> \u003cp>So, z is least for tanØ = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>As, Sec\u003csup>2\u003c/sup>Ø = 1 + tan\u003csup>2\u003c/sup>Ø = 1 + \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>secØ = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And tan\u003csup>2\u003c/sup>Ø = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n θ\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>cot\u003csup>2\u003c/sup>Ø = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And cosec\u003csup>2\u003c/sup>Ø = 1 + cot\u003csup>2\u003c/sup>Ø = 1 + \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>cosecØ = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n b\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089330php4Ht7A8_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"383\" height=\"128\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089330php4Ht7A8.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 383 128'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q115. Find the points at which the function f given by f (x) = (x - 2)\u003csup>4\u003c/sup> (x + 1)\u003csup>3\u003c/sup> has (i) local maxima (ii) local minima (iii) point of inflexion\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.13. \u003c/strong>We have,\u003c/p> \u003cp>f(x) = (\u003cem>x\u003c/em>- 2)\u003csup>4\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup>.\u003c/p> \u003cp>So, f(x) = (\u003cem>x\u003c/em>- 2)\u003csup>4\u003c/sup>. 3(\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> + (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup>. 4(\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup>.\u003c/p> \u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> [\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> [3 (\u003cem>x\u003c/em>- 2) + 4(\u003cem>x\u003c/em> + 1)]\u003c/p> \u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (3\u003cem>x\u003c/em>- 6 + 4\u003cem>x\u003c/em> + 4)\u003c/p> \u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (7\u003cem>x\u003c/em>- 2).\u003c/p> \u003cp>At f(x) = 0.\u003c/p> \u003cp>(\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>. (7\u003cem>x\u003c/em>- 2) = 0.\u003c/p> \u003cp>x = 2, x = -1 or x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>As (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>> 0,we shave evaluate for the remaining factor.\u003c/p> \u003cp>At x = 2,\u003c/p> \u003cp>When x\u003c 2, f(x) = ( -ve)(+ ve)(+ ve) = ( -ve) \u003c 0.\u003c/p> \u003cp>When x> 2, f(x) = (+ ve)(+ ve)(+ ve) = (+ ve) > 0.\u003c/p> \u003cp>Øf(x) change from ( -ve) to (+ ve) as x increases\u003c/p> \u003cp>So, x = 2 is a point of local minima\u003c/p> \u003cp>At x = -1.\u003c/p> \u003cp>When x\u003c -1, f(x) = ( -ve)(+ ve)( -ve) =\notin , ve > 0.\u003c/p> \u003cp>When x> -1, f(x) = ( -ve)(+ ve)(+ ve) =\notin , ve > 0.\u003c/p> \u003cp>So, f(x) does not change through x ---1.\u003c/p> \u003cp>Hence, x = -1 is a point of infixion\u003c/p> \u003cp>At x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>When x\u003c \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = ( -ve)(+ ve)( -ve) = ( -ve > 0.\u003c/p> \u003cp>When x\u003c \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> f(x) = ( -ve)(+ ve)(+ ve) = ( -ve \u003c 0.\u003c/p> \u003cp>f(x) change from (+ ve) to ( -ve) as x increases through \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> is a point of local maxima\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q116. Find the absolute maximum and minimum values of the function f given by f (x) =cos\u003csup>2\u003c/sup> x + sin x, x \in [0, π]\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.14. \u003c/strong>We have,\u003c/p> \u003cp>f(x) = cos\u003csup>2\u003c/sup>x + sin x, x\in [0,π ].\u003c/p> \u003cp>So, f(x) = 2 cos x ( -sin x) + cos x = cos x (1 - 2 sin x).\u003c/p> \u003cp>At f(x) = 0\u003c/p> \u003cp>cosx (1 - 2 sin x) = 0\u003c/p> \u003cp>cosx = 0 or 1 - 2 sin x = 0\u003c/p> \u003cp>cosx = cos \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> or sin x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = sin \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = sin \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>x= \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> , x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> and x = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> [0,π ].\u003c/p> \u003cp>So, f \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> = cos\u003csup>2\u003c/sup>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> + sin \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> = 1.\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089825phpXz99UG_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"308\" height=\"145\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734089825phpXz99UG.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 308 145'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>Absolute minimum of f(x) = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 5\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> and absolute minimum of f(x) = 1.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q117. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.15. \u003c/strong>Let x be the radius of the sphere ad x be radius of the right circular cone.\u003c/p> \u003cp>Let height of cone = y\u003c/p> \u003cp>Then, in ΔOBA,\u003c/p> \u003cp>(\u003cem>y\u003c/em>-\u003cem>r\u003c/em>)\u003csup>2\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = \u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cem>y\u003c/em>\u003csup>2\u003c/sup> + \u003cem>r\u003c/em>\u003csup>2\u003c/sup>- 2\u003cem>ry\u003c/em> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = \u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup>=2\u003cem>ry \u003c/em>-\u003cem> y\u003c/em>\u003csup>2\u003c/sup>\u003c/p> \u003cp>So, the volume V of the cone is\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n V=\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \cdot\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n {\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmi>\n s\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \timeshight\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n }\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \times\n \u003c/mo>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>So, \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>And \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>At \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n V\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmrow>\n \u003cmo>\n [\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>4\u003cem>x \u003c/em>-\u003cem>y\u003c/em>- 3\u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> asy> 0.\u003c/p> \u003cp>At y = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n [\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \u003c\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Ø V is maximum when y = \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q.118. Let f be a function defined on [a, b] such that f '(x) > 0, for all x \in (a, b).\u003c/strong>\u003c/p> \u003cp>\u003cstrong>Then prove that f is an increasing function on (a, b).\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.16. \u003c/strong>We have,\u003c/p> \u003cp>f(x) defined on [a, b]\u003c/p> \u003cp>And f(x) > 0 \forall x \in [a, b].\u003c/p> \u003cp>Let \u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2 \u003c/sub>\in [a, b] and \u003cem>x\u003c/em>\u003csub>2\u003c/sub>>\u003cem>x\u003c/em>\u003csub>1\u003c/sub>\u003c/p> \u003cp>In the internal \u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2\u003c/sub>], f(x) will also be continuous and differentiable.\u003c/p> \u003cp>Hence by mean value theorem, there exist c [\u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2\u003c/sub>] such that\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n c\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>f(x) > 0 \forall x \in [a, b].\u003c/p> \u003cp>Then, f(c) > 0.\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n >\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>i.e., f(\u003cem>x\u003c/em>\u003csub>2\u003c/sub>) -\u003cem>f\u003c/em>(\u003cem>x\u003c/em>\u003csub>1\u003c/sub>) > 0\u003c/p> \u003cp>f(\u003cem>x\u003c/em>\u003csub>2\u003c/sub>) >\u003cem>f\u003c/em>(\u003cem>x\u003c/em>\u003csub>1\u003c/sub>).\u003c/p> \u003cp>Hence, the function f(x) is always increasing on [a, b]\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q119. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/\sqrt3. Also find the maximum volume.\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1742280589phpkbwJtE_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"421\" height=\"472\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1742280589phpkbwJtE.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 421 472'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003cp>\n \u003cpicture>\n \u003cimg alt=\"\" width=\"292\" height=\"126\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093061phpcHOQmp.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 292 126'>\u003c/svg>\">\n \u003c/picture>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q120. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n t\n \u003c/mi>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n α\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.18.\u003c/strong>\u003c/p> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093179php9W4ar2_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"480\" height=\"454\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093179php9W4ar2.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 480 454'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093194phpKMnn8r_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"261\" height=\"97\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093194phpKMnn8r.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 261 97'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \n \u003cdiv class=\"photo-widget-full\"> \n \u003cdiv class=\"figure\">\n \u003cpicture>\n \u003csource media=\"(max-width: 500px)\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093208phpNsNZiV_480x360.jpeg\">\n \u003cimg alt=\"\" width=\"348\" height=\"442\" data-original=\"https://images.shiksha.com/mediadata/images/articles/1734093208phpNsNZiV.jpeg\" class=\"lazy\" src=\"data:image/svg+xml,\u003csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 348 442'>\u003c/svg>\">\n \u003c/picture>\n \u003c/div> \n \u003c/div> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Choose the correct answer in the questions from 19 to 24.\u003c/strong>\u003c/p> \u003cp> \u003c/p> \u003cp>\u003cstrong>Q121. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.19. \u003c/strong>Let x be the depth of the wheat inside the cylindrical tank is with radius = 10 cm\u003c/p> \u003cp>Then, Volume V of the cylindrical tank is\u003c/p> \u003cp>V = π(10)\u003csup>2\u003c/sup>\u003cem>x\u003c/em> = 100πp\u003cem>x\u003c/em> m\u003csup>3\u003c/sup>\u003c/p> \u003cp>As \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmtext>\n \n \u003c/mtext>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n t\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmfrac bevelled=\"true\">\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> i e, rate of increasing of depth\u003c/p> \u003cp>option (A) is correct\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q122. The line y = mx + 1 is a tangent to the curve y\u003csup>2\u003c/sup> = 4x if the value of m is\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(A)\u003c/strong> 1 \u003cstrong>(B)\u003c/strong> 2 \u003cstrong>(C)\u003c/strong> 3 \u003cstrong>(D)\u003c/strong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.21. \u003c/strong>The equation of the tangent to the given curve is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, substituting \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> in \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> we get:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Since a tangent touches the curve at one point, the roots of equation (i) must be equal.\u003c/p> \u003cp>Therefore, we have:\u003c/p> \u003cp>Discriminant = 0\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n m\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the required value of m is 1.\u003c/p> \u003cp>Therefore, option (A) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q123. The normal at the point (1,1) on the curve 2y + x\u003csup>2\u003c/sup> = 3 is (A) x + y = 0 (B) x - y = 0 (C) x + y +1 = 0 (D) x - y = 1\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.22. \u003c/strong>The equation of the given curve is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/p> \u003cp>Differentiate with respect to x, we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The slope of the normal to the given curve at point (1,1) is\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the equation of the normal to the given curve at (1,1) is given as:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Therefore, option (B) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q124. The normal to the curve x\u003csup>2\u003c/sup> = 4y passing (1,2) is (A) x + y = 3 (B) x - y = 3 (C) x + y = 1 (D) x - y = 1\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.23. \u003c/strong>The equation of the given curve is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Differentiating with respect to x, we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The slope of the normal to the given curve at point (h,k) is given by,\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> Equation of the normal at point (h,k) is given as:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Now, it is given that the normal passes through the point (1,2).\u003c/p> \u003cp>Therefore, we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmi>\n o\n \u003c/mi>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Since (h,k) lies on the curve \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span> ,we have \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n k\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>From equation (i), we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n k\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the equation of the normal is given as:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Therefore, option (A) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\"> \u003cp>\u003cstrong>Q125. The points on the curve 9y\u003csup>2\u003c/sup> = x\u003csup>3\u003c/sup> , where the normal to the curve makes equal intercepts with the axes are\u003c/strong>\u003c/p> \u003cp>\u003cstrong>(A) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> (B) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> (C) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span> (D) \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/p> \u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\"> \u003cp>\u003cstrong>A.24. \u003c/strong>The equation of the given curve is \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Differentiate with respect to x, we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>The slope of the normal to the given curve at point \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> is\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n ]\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span> The equation of the normal to the curve at \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> is\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n -\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n y\n \u003c/mi>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>It is given that the normal makes intercepts with the axes.\u003c/p> \u003cp>Therefore, we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n ∴\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Also, the point \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> lies on the curve, so we have\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo>\n .\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>From (i) and (ii), we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>From (ii), we have:\u003c/p> \u003cp>\u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmtable columnalign=\"left\">\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003cmtr>\n \u003cmtd>\n \u003cmrow>\n \u003cmo>\n \Rightarrow\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n y\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msub>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/mtd>\n \u003c/mtr>\n \u003c/mtable>\n \u003c/math>\u003c/span>\u003c/p> \u003cp>Hence, the required points are \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ,\n \u003c/mo>\n \u003cmo>\n \pm\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 8\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span> .\u003c/p> \u003cp>Therefore, option (A) is correct.\u003c/p> \u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cp> \u003c/p>","order":9,"subSectionId":"71953","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"180138","status":"live","updateDateOn":false,"updatedOn":"2025-06-03 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Overview of Applications of Derivatives - Class 12 Maths","h2Id":"content_toc_952240"},{"h2Text":"Application of Derivatives NCERT Solution PDF: Free Download","h2Id":"content_toc_902430"},{"h2Text":"Class 12 Maths Application of Derivatives: Key Topics, Weightage ","h2Id":"content_toc_902428"},{"h2Text":"Important Formulas of Application of Derivatives Class 12","h2Id":"content_toc_952250"},{"h2Text":"Class 12 Application of Derivatives Exercise 6.1 Solutions","h2Id":"content_toc_911042"},{"h2Text":"NCERT Solutions for Class 12 Maths Ch 6 Application of Derivatives- FAQs","h2Id":"content_toc_893002"},{"h2Text":"Class 12 Application of Derivatives 6.5 Solutions- (Old NCERT)","h2Id":"content_toc_910806"},{"h2Text":"NCERT Class 12 Application of Derivatives Miscellaneous Exercise Solutions","h2Id":"content_toc_910826"}],"pteTopExamWidget":null,"updatedOn":"2025-08-04 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Use them for practice and improve your speed and accuracy. You can download Maths Ncert Solutions class 12th previous year question papers and the preparation guides for free.","importantdates":"Get all Maths Ncert Solutions class 12th 2023 important dates such as registration date, application date, admit card date, exam date, result date and counselling date. Sign up so that you don’t miss a single exam date or an update on Maths Ncert Solutions class 12th 2023 exam."},"dfpAdUnits":null,"ctpURL":{"url":"/search?q=Maths Ncert Solutions class 12th&s[]=25&ed[]=et_20::dm_39&ex[]=14833&pn=1&rf=examPage&apk=true&skr=false&rURL=false&pt=search_page","urlHost":"national","absoluteUrl":"https://www.shiksha.com/search?q=Maths Ncert Solutions class 12th&s[]=25&ed[]=et_20::dm_39&ex[]=14833&pn=1&rf=examPage&apk=true&skr=false&rURL=false&pt=search_page"},"ctpCount":0,"faq":{"@context":"https://schema.org","@type":"FAQPage","mainEntity":[{"@type":"Question","name":"27. Prove that y=4sinθ(2+cosθ)-θ is an increasing function of θ (0,π2) .","acceptedAnswer":{"@type":"Answer","text":"We have, y = 4sinθ(2+cosθ)-θ Differentiating w rt.Ø we get, dydθ=ddθ[4sinθ2+cosθ-θ] =(2+corθ)ddθ(4sinθ)-4sinθddθ(2+corθ)(2+cosθ)2-dθdθ. =(2+cosθ)(4cosθ)-4sinθ(-sinθ)(2+cosθ)2-1 =8cosθ+4cos2θ+4sin2θ\cdot(2+cosθ)2.-1 =8cosθ+4(cos2θ+sin2θ)-(2+cosθ)2(2+cosθ)2 =8corθ+4-[4+cos2θ+4cosθ](2+corθ)2. =8cosθ+4-4-cos2θ-4cosθ(2+cosθ)2 =4cosθ-cos2θ(2+cosθ)2=cosθ(4-cosθ)(2+cosθ)2 When θ\in[0,π2] we know that, 0\leqcosθ\leq1. So, 4-cosθ>0 And also, (2 + cos θ)2> 0. ∴dydθ\geq0 \forallθ\in[0,π2] Hence, y is an increasing fxn of θ in [0,π2]"}},{"@type":"Question","name":"74.Find the maximum and minimum values, if any, of the following functions given by. (i) f(x)=(2x-1)2+3","acceptedAnswer":{"@type":"Answer","text":"(i) We have, f(x) = (2x - 1)2 + 3. For all x\in?, (2x-1)2\geq0 (2x - 1)2 + 3 \geq 3. f(x) \geq 3. ∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = 12 Again as x\to\infty,f(x)\to\infty as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values. (ii) f(x)=9x2+12x+2 A(ii) We have, f(x) = 92 + 12x + 2. f(x)=9[x2+12x9+29] (Taking 9 common from each team). f(x)=9[x2+4x3+29] f(x)=9[x2+2\times2x3+(23)2-(23)2+29] f(x)=9[(x+23)2-49+29] f(x)=9[\cdot(x+23)2-29]=9(x+23)2-2 For all x\in?,(x+23)2\geq0 \Rightarrow(x+23)2-2\geq\cdot 0-2 \Rightarrow f(x)\geq - 2. ∴The minimum value of f(x) = -2 when x+23=0 And as x\to\infty,f(x)\to\infty so f(x) has \Rightarrowx=-23. no maximum values. (iii) f(x) = (x - 1)2 + 10 A(iii) we have, f(x) = - (x - 1)2 + 10 For all x\in?, (x-1)2\geq0. \Rightarrow (x - 1)2 \leq 0 \Rightarrow -(x- 1)2 + 10 \leq 10. \Rightarrow f(x) \leq 10. ∴maximum value of f(x) = 10 when x - 1 = 0 \Rightarrow x = 1. And minimum value of f(x) does not exist. (iv) g(x) = x3 + 1 A(iv) we have, g(x) = x3 + 1 For the given fxn, x\in?, g(x)\to\infty as x\to\infty and g(x)-\infty as x\to-\infty Q Maximum and minimum value does not exist."}},{"@type":"Question","name":"14. Sand is pouring from a pipe at the rate of 12 cm3 /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?","acceptedAnswer":{"@type":"Answer","text":"Let r cm and h cm be the radius and the height of the cone. Then, h = 16 r. H = 6h So, volume, V of the cone = 13 πr2h =13π(6h)2\timesh. 4ddt. = 12 \times h3 Rate of change of volume of the cone wrt the height is dVdh=ddh(12πh3) = 12 \times π \times 3 \times h2. As the sand is pouring from the pipe at rate of 12 cm35 we have ddt=12 \Rightarrowdvdh\timesdhdt=12 \Rightarrow36πh2dhdt=12. \Rightarrow dhdt=1236πh2=13πh2. ∴dhdt|h=4 =13π\times(4)2=148π. Hence, the height is increasing at the rate of 148x cm/s."}},{"@type":"Question","name":"63. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is (A) 3 (B) 13 (C) -3 (D) - 13","acceptedAnswer":{"@type":"Answer","text":"Given, y=2x2+3sinx Slope of tangent, dydx=4x+3cosx dydx|x=0=4\times0+3\timescos0=3 So, slope of normal =-13 Option (D) is correct."}},{"@type":"Question","name":"99. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin-113 .","acceptedAnswer":{"@type":"Answer","text":"Kindly go through the solution"}}]},"prepFaq":{"name":"Quiz on Application of Derivatives","typicalAgeRange":"13-20","assesses":"Application of Derivatives","educationalLevel":"Class 12th","educationalAlignment":[{"alignmentType":"educationalSubject","targetName":"Maths","@type":"AlignmentObject"},{"alignmentType":"educationalSubject","targetName":"Application of Derivatives","@type":"AlignmentObject"}],"about":{"@type":"Thing","name":"Maths Ncert Solutions class 12th"},"hasPart":[{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>10\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>11\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>12\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>9\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>Surface area, S = 4pr\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmo>.\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>r\u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003cmi>? \u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>t\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814779295\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>? \u003c/mo> \u003cmi>s\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814779296\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>Initially t = 0, r = 3\u003c/p>\u003cp>c = 36 p\u003c/p>\u003cp>When t = 5, r = 7, k = 32p\u003c/p>\u003cp>When t = 9, r = r, r = 9\u003c/p>\u003cp> \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> 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\u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> 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\u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>Let f(x) = \u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math> Then which of the following statements are true?\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>P : x = 0 is a point of local minima of f\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Q : x = \u003cmath>\u003cmrow>\u003cmroot>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/mroot>\u003c/mrow>\u003c/math> is a point of inflection of f\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>R : \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/math> is increasing for x > \u003cmath>\u003cmrow>\u003cmroot>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/mroot>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cstrong>Only P and Q\u003c/strong>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cstrong>Only P and R\u003c/strong>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cstrong>Only Q and R\u003c/strong>\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>\u003cstrong>All P, Q and R\u003c/strong>\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmi mathvariant=\"script\">l\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>From graph : P, Q, R\u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>The number of points, where the function f : R R,\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math> is NOT differentiable, is:\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>1\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>3\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>4\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>2\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>x = 1, 4 (doubtful points)\u003c/p>\u003cp>Diff. at x = 1\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>R\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>L\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmo>}\u003c/mo>\u003c/mrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math> Not diff. at x = 4\u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>The sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>/\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>The sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>/\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>Let f : R® R be a continuous function satisfying f(x) + f(x + k) = n, for all x \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\in\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> R where k > 0 and n is a positive integer. If \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>t\u003c/mi> \u003cmi>h\u003c/mi> \u003cmi>e\u003c/mi> \u003cmi>n\u003c/mi> \u003cmo>:\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> \u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 4nk\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 2nk\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp> I\u003csub>1\u003c/sub> + nI\u003csub>2\u003c/sub> = 6n\u003csup>2\u003c/sup>k\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp> I\u003csub>1\u003c/sub>+ nI\u003csub>2\u003c/sub> = 4n\u003csup>2\u003c/sup>k\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>\forall\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>\in\u003c/mo> \u003cmi>R\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmo>&\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>Replace x by x + k. \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>From (i) & (ii), f(x + 2k) = f(x).\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  is periodic with period = 2k.\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>put x = t + k\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>n\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>n\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmi>k\u003c/mi> \u003cmo>.\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>  \u003c/p>\u003cp>  \u003c/p>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>"},"@type":"Answer"}],"@type":"Question"},{"eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","encodingFormat":"text/html","name":"Practice Problem on Application of Derivatives","text":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","suggestedAnswer":[{"position":0,"encodingFormat":"text/html","text":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","@type":"Answer"},{"position":0,"encodingFormat":"text/html","text":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","@type":"Answer"}],"acceptedAnswer":[{"position":1,"encodingFormat":"text/html","text":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","answerExplanation":{"@type":"Comment","text":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>"},"@type":"Answer"}],"@type":"Question"}],"@context":"https://schema.org","@type":"Quiz"},"sectionsInfo":{"homepage":{"h1":"Class 12 Maths Applications of Derivatives - Step-by-Step NCERT Solutions","viewCount":973,"h2":"Maths Ncert Solutions class 12th 2023 Dates","h2Description":"Take a look at the table below for the complete schedule of Maths Ncert Solutions class 12th 2023 exam","guideUrl":"/mediadata/examGuides/examSectionGuide4569_Overview_2025-09-13-03-31.pdf","firstImageUrl":"/mediadata/images/articles/1734068563phpiAlP6W.jpeg","urlMapping":{"l2Url":"/preparation/maths-ncert-solutions-class-12th-application-of-derivatives-4569-ns","l3Urls":null,"l3ToL4Urls":null}}},"officialWebsiteWithUTM":null,"firstImageInContent":null,"examPhase":"default_phase","prePostLabel":"Both","examMetaDataDTO":{"phoneNo":null,"officialWebsite":null,"isERWidget":false},"ctaInfoData":{"useEditorialCTA":true,"ctaInfoList":[],"examPhase":null},"class_9":null,"class_10":null,"class_11":null,"class_12":null,"questionPapersAvailable":false,"futureDateAvailable":false,"pattern":null,"syllabus":null,"results":null,"cutoff":null,"admitcard":null,"answerkey":null,"applicationform":null,"counselling":null,"samplepapers":{"wiki":null,"files":null,"guideUrls":null,"sectionalData":{"sectionName":null,"entityType":null,"contentData":[],"childPageDataConfig":{"homepage":{"categoryAllowed":{"intro":1,"content":30,"contactus":1},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null},"l4ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null}}},"tocList":null,"pteTopExamWidget":null,"updatedOn":""}},"slotbooking":null,"vacancies":null,"news":null,"callletter":null,"preptips":null,"sectionname":["homepage"],"l3Mapping":{},"l4Mapping":{}},"statusCode":200,"urlHash":"eyJ1cmwiOiIvcHJlcGFyYXRpb24vbWF0aHMtbmNlcnQtc29sdXRpb25zLWNsYXNzLTEydGgtYXBwbGljYXRpb24tb2YtZGVyaXZhdGl2ZXMtNDU2OS1ucyIsInVzZU9sZEZvcm1hdCI6ZmFsc2UsInNob3dTaW1pbGFyRXhhbXMiOmZhbHNlfQ==","url":"/preparation/maths-ncert-solutions-class-12th-ns","activeSection":"homepage","childPageHelper":{"id":null,"childPageName":"homepage","childPageType":"L2","l2Name":null,"l3Name":null,"uniqueChildPageName":"homepage","displayChildPageName":null,"displayParentPageName":null,"childPageUrl":"/preparation/maths-ncert-solutions-class-12th-application-of-derivatives-4569-ns"},"breadCrumb":[{"name":"Home","url":"/","linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":true},{"name":"Preparation","url":null,"linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":false},{"name":"Maths Ncert Solutions class 12th","url":null,"linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":false}],"articleData":null,"featuredColleges":null,"similarExams":{"examTuple":null,"groupLevel":null,"pageNum":null,"totalPages":0},"examPhaseData":null,"epUpdates":null,"odnDetailsListResponseDTO":{"odnDetailsListResponse":{"odnRedAlert":null,"odnAlertList":null},"haveMoreTuples":false},"seoData":{"metaTitle":"Class 12 Maths Applications of Derivatives - Step-by-Step NCERT Solutions","metaDescription":"Explore Class 12 Maths Chapter 6 - Applications of Derivatives with complete NCERT Solutions. 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sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-sum-of-absolute-maximum-and-absolute-minimum-values-of-the-function-f-x-2x-lt-sup-gt-qna-11692028"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7756,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>/\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7758,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":true},{"id":7760,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7762,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135552,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180458","questionId":11687542,"answerId":11687574,"questionText":"\u003cp>\u003cstrong>Let f : R® R be a continuous function satisfying f(x) + f(x + k) = n, for all x \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\in\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> R where k > 0 and n is a positive integer. If \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>t\u003c/mi> \u003cmi>h\u003c/mi> \u003cmi>e\u003c/mi> \u003cmi>n\u003c/mi> \u003cmo>:\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> \u003c/strong>\u003c/p>","answerText":"\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>\forall\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>\in\u003c/mo> \u003cmi>R\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmo>&\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>Replace x by x + k. \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>From (i) & (ii), f(x + 2k) = f(x).\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  is periodic with period = 2k.\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>put x = t + k\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>n\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>n\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmi>k\u003c/mi> \u003cmo>.\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>  \u003c/p>\u003cp>  \u003c/p>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-let-f-r-reg-r-be-a-continuous-function-satisfying-f-x-f-x-k-n-for-all-x-lt-strong-gt-lt-span-qna-11687542"},"anaFaqSource":"manual","generationSource":"manual","templateId":141,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7204,"optionText":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 4nk\u003c/p>","order":0,"correct":false},{"id":7206,"optionText":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 2nk\u003c/p>","order":1,"correct":false},{"id":7208,"optionText":"\u003cp> I\u003csub>1\u003c/sub>+ nI\u003csub>2\u003c/sub> = 4n\u003csup>2\u003c/sup>k\u003c/p>","order":2,"correct":true},{"id":7210,"optionText":"\u003cp> I\u003csub>1\u003c/sub> + nI\u003csub>2\u003c/sub> = 6n\u003csup>2\u003c/sup>k\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135554,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180458","questionId":11638756,"answerId":11638760,"questionText":"\u003cp>\u003cstrong>The number of points, where the function f : R R,\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math> is NOT differentiable, is:\u003c/strong>\u003c/p>","answerText":"\u003cp> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>x = 1, 4 (doubtful points)\u003c/p>\u003cp>Diff. at x = 1\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>R\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>L\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmo>}\u003c/mo>\u003c/mrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math> Not diff. at x = 4\u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-number-of-points-where-the-function-f-r-r-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-lt-qna-11638756"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":1894,"optionText":"\u003cp>1\u003c/p>","order":0,"correct":false},{"id":1896,"optionText":"\u003cp>2\u003c/p>","order":1,"correct":true},{"id":1898,"optionText":"\u003cp>3\u003c/p>","order":2,"correct":false},{"id":1900,"optionText":"\u003cp>4\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true}],"180460":[{"anaFaqId":49226530,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11691928,"answerId":11691952,"questionText":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","answerText":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-for-the-function-f-x-4-log-lt-sub-gt-e-lt-sub-gt-x-ndash-1-ndash-2x-lt-sup-gt-2-lt-sup-gt-qna-11691928"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7740,"optionText":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","order":0,"correct":false},{"id":7742,"optionText":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","order":1,"correct":false},{"id":7744,"optionText":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","order":2,"correct":true},{"id":7746,"optionText":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":49226532,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11691968,"answerId":11692008,"questionText":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-if-the-tangent-at-the-point-x-lt-sub-gt-1-lt-sub-gt-y-lt-sub-gt-1-lt-sub-gt-on-the-curve-qna-11691968"},"anaFaqSource":"manual","generationSource":"manual","templateId":123,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7748,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7750,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":false},{"id":7752,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7754,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":true}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":49226534,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11692028,"answerId":11692060,"questionText":"\u003cp>\u003cstrong>The sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> 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\u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-sum-of-absolute-maximum-and-absolute-minimum-values-of-the-function-f-x-2x-lt-sup-gt-qna-11692028"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7756,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> 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\u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":true},{"id":7760,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> 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If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :\u003c/strong>\u003c/p>","answerText":"\u003cp>Surface area, S = 4pr\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmo>.\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>r\u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003cmi>? \u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>t\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814779295\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>? \u003c/mo> \u003cmi>s\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814779296\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>Initially t = 0, r = 3\u003c/p>\u003cp>c = 36 p\u003c/p>\u003cp>When t = 5, r = 7, k = 32p\u003c/p>\u003cp>When t = 9, r = r, r = 9\u003c/p>\u003cp> \u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-surface-area-of-a-balloon-of-spherical-shape-being-inflated-increases-at-a-constant-rate-qna-11691674"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7684,"optionText":"\u003cp>9\u003c/p>","order":0,"correct":true},{"id":7686,"optionText":"\u003cp>10\u003c/p>","order":1,"correct":false},{"id":7688,"optionText":"\u003cp>11\u003c/p>","order":2,"correct":false},{"id":7690,"optionText":"\u003cp>12\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135546,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210580","questionId":11691928,"answerId":11691952,"questionText":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","answerText":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-for-the-function-f-x-4-log-lt-sub-gt-e-lt-sub-gt-x-ndash-1-ndash-2x-lt-sup-gt-2-lt-sup-gt-qna-11691928"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7740,"optionText":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","order":0,"correct":false},{"id":7742,"optionText":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","order":1,"correct":false},{"id":7744,"optionText":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","order":2,"correct":true},{"id":7746,"optionText":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135548,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210580","questionId":11691968,"answerId":11692008,"questionText":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-if-the-tangent-at-the-point-x-lt-sub-gt-1-lt-sub-gt-y-lt-sub-gt-1-lt-sub-gt-on-the-curve-qna-11691968"},"anaFaqSource":"manual","generationSource":"manual","templateId":123,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7748,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7750,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":false},{"id":7752,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7754,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":true}]},"experiment":false,"isAIGenerated":false,"crawlable":true}],"210582":[{"anaFaqId":53135542,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210582","questionId":11638812,"answerId":11638818,"questionText":"\u003cp>\u003cstrong>Let f(x) = \u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math> Then which of the following statements are true?\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>P : x = 0 is a point of local minima of f\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Q : x = \u003cmath>\u003cmrow>\u003cmroot>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/mroot>\u003c/mrow>\u003c/math> is a point of inflection of f\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>R : \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/math> is increasing for x > \u003cmath>\u003cmrow>\u003cmroot>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/mroot>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmi mathvariant=\"script\">l\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>From graph : P, Q, R\u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-let-f-x-amp-nbsp-lt-math-gt-lt-mrow-gt-lt-msup-gt-lt-mrow-gt-lt-mn-gt-3-lt-mn-gt-lt-mrow-qna-11638812"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":1918,"optionText":"\u003cp>\u003cstrong>Only P and Q\u003c/strong>\u003c/p>","order":0,"correct":false},{"id":1920,"optionText":"\u003cp>\u003cstrong>Only P and R\u003c/strong>\u003c/p>","order":1,"correct":false},{"id":1922,"optionText":"\u003cp>\u003cstrong>Only Q and R\u003c/strong>\u003c/p>","order":2,"correct":false},{"id":1924,"optionText":"\u003cp>\u003cstrong>All P, Q and 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Check here the list of top NITs with their MHRD-NIRF rankings, courses offered, total seat intake and cut offs.","lastModifiedDate":"2025-09-04 14:50:43","views":3594999,"userId":29781649,"comments":205,"shares":313,"discussionTopicId":5232713,"tagType":"exam","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":117273,"url":"/boards/articles/cbse-10th-maths-question-paper-pdf-download-blogId-117273","title":"CBSE 10th Maths Question Paper 2025, 2024, 2023, 2022, 2020, 2019 PDF Download","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1709092170phpNR1yFa_b.jpeg","summary":"CBSE class 10 Maths question paper PDF 2025 is available here. Students can also download CBSE 10th Maths previous year question paper PDF to score well in class 10th board exam. Download CBSE 10th mathematics question paper 2025 along with PYQ of 2024, 2023, 2022, 2020 and 2019 here.","lastModifiedDate":"2025-09-13 14:22:03","views":3428198,"userId":65362629,"comments":0,"shares":4955,"discussionTopicId":7294495,"tagType":"multi","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":28509,"url":"/medicine-health-sciences/articles/neet-chapter-wise-weightage-blogId-28509","title":"NEET Chapter Wise Weightage 2026 PDF by NTA: Physics, Chemistry & Biology","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1743767698phpKlzK7b_b.jpeg","summary":"NEET Chapter Wise Weightage 2026 - Knowing the weightage of chapters in NEET has helped candidates excel in the examination. Check NEET chapter-wise weightage PDF for Physics, Chemistry, and Biology here. Knowing the chapter-wise weightage will help you prepare a rock solid preparation strategy for the exam day.","lastModifiedDate":"2025-10-08 18:32:53","views":3110280,"userId":18325315,"comments":115,"shares":4607,"discussionTopicId":5853393,"tagType":"exam","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":12884,"url":"/engineering/articles/iits-in-india-blogId-12884","title":"List of Top IITs in India 2025: NIRF Ranking, Courses, Seats, Fees & Placement","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1568101678phpqodvPW_b.jpeg","summary":"Check out the ranking, seats and cut offs of all 23 IITs including IIT Kharagpur, IIT Bombay, IIT Madras, IIT Kanpur, IIT Delhi, IIT Guwahati, IIT Roorkee, IIT Ropar, IIT Bhubaneswar, IIT Gandhinagar, IIT Hyderabad etc. NIRF ranking 2025 for engineering has been released. Eight of the 23 IITs feature in the Top 10 of the NIRF engineering ranking.","lastModifiedDate":"2025-09-04 14:57:03","views":2560201,"userId":29781649,"comments":41,"shares":432,"discussionTopicId":3725168,"tagType":"multi","liveBlogStartDate":null,"liveBlogEndDate":null}],"latest":[{"id":211980,"url":"/news/boards-hbse-date-sheet-2026-class-12-10-bseh-org-live-updates-blogId-211980","title":"HBSE Date Sheet 2026 Class 12, 10 @bseh.org Live Updates","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760423559phpaNAcWg_b.jpeg","summary":"BSEH is all set to release the HBSE Date Sheet 2026 Class 10, 12 soon on the official website- bseh.org. Students preparing for their Haryana Board Exams 2026 Class 10, 12 can check the exam dates here as soon as they are made available. HBSE is expected to begin the Class 10, 12 board exams from February 20, 2026. Check live updates here...","lastModifiedDate":"2025-10-19 01:03:00","views":0,"userId":13844819,"comments":0,"shares":0,"discussionTopicId":12418980,"tagType":"multi","liveBlogStartDate":"2025-10-14 12:04:00","liveBlogEndDate":"2025-10-20 12:04:00"},{"id":212002,"url":"/news/boards-board-exam-date-sheet-2026-class-10th-12th-live-updates-blogId-212002","title":"Board Exam Date Sheet 2026 Class 10th, 12th Live Updates; ICSE, Maharashtra, Tamil Nadu, GSEB Time Table Soon","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760434438php1GCbjT_b.jpeg","summary":"Board exam date sheet 2026 for classes 10th and 12th for various states will be released soon. ICSE and ISC time table 2025 is awaited, however, tentative CBSE 10, 12 datesheet 2026 has already been released. Check live updates on board exam date sheet 2026 here. Board exam date sheet 2026 for classes 10th and 12th for various states will be releas","lastModifiedDate":"2025-10-19 00:43:00","views":0,"userId":59961703,"comments":1,"shares":8,"discussionTopicId":12419212,"tagType":"multi","liveBlogStartDate":"2025-10-14 15:10:00","liveBlogEndDate":"2025-10-21 23:59:00"}]},"allExamPageGuide":null,"pageSchema":{"@context":"https://schema.org","@type":"Article","inLanguage":"en","headline":"Class 12 Maths Applications of Derivatives - Step-by-Step NCERT Solutions","description":"Explore Class 12 Maths Chapter 6 - Applications of Derivatives with complete NCERT Solutions. 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Prove that \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> is an increasing function of θ \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Differentiating w rt.Ø we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>θ\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> we know that, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And also, (2 + cos\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>)\u003csup>2\u003c/sup>> 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmo>\forall\u003c/mo>\u003cmi>θ\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, y is an increasing \u003cem>fx\u003c/em>\u003csup>n\u003c/sup> of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-27-prove-that-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-mi-qna-11425044"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398512,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426800,"answerId":11426810,"questionText":"\u003cp>\u003cstrong>74.Find the maximum and minimum values, if any, of the following functions\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>given by.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>We have, f(x) = (2\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3.\u003c/p>\u003cp>For all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(2\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3 \geq 3.\u003c/p>\u003cp>f(x) \geq 3.\u003c/p>\u003cp>∴The minimum value of f(x) = 3. When 2x - 1 = 0--> x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Again as \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> as there is vouppa bound to ‘x’ value hence, f(x) has no maximum values.\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(ii)\u003c/p>\u003cp>We have, f(x) = 9\u003csup>2\u003c/sup> + 12x + 2.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> (Taking 9 common from each team).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>\cdot\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>For all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\geq\u003c/mo>\u003cmo>\cdot\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x)\geq - 2.\u003c/p>\u003cp>∴The minimum value of f(x) = -2 when \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And as \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> so f(x) has \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>no maximum values.\u003c/p>\u003cp>\u003cstrong>(iii)\u003c/strong> f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 10\u003c/p>\u003cp>A(iii)\u003c/p>\u003cp>we have, f(x) = - (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 10\u003c/p>\u003cp>For all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> \leq 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -(\u003cem>x\u003c/em>- 1)\u003csup>2\u003c/sup> + 10 \leq 10.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) \leq 10.\u003c/p>\u003cp>∴maximum value of f(x) = 10 when x - 1 = 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 1.\u003c/p>\u003cp>And minimum value of f(x) does not exist.\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 1\u003c/p>\u003cp>A(iv)\u003c/p>\u003cp>we have, g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 1\u003c/p>\u003cp>For the given \u003cem>fx\u003c/em>\u003csup>n\u003c/sup>, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Q Maximum and minimum value does not exist.\u003c/p>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-74-find-the-maximum-and-minimum-values-if-any-of-the-following-functions-lt-strong-gt-lt-qna-11426800"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398392,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424850,"answerId":11424854,"questionText":"\u003cp>\u003cstrong>14. Sand is pouring from a pipe at the rate of 12 cm\u003csup>3\u003c/sup> /s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r cm and h cm be the radius and the height of the cone. Then,\u003c/p>\u003cp>h =\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> r. H = 6h\u003c/p>\u003cp>So, volume, V of the cone =\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003cem>h\u003c/em>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\times\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 12 \times \u003cem>h\u003c/em>\u003csup>3\u003c/sup>\u003c/p>\u003cp>Rate of change of volume of the cone wrt the height is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> = 12 \times π \times 3 \times \u003cem>h\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>As the sand is pouring from the pipe at rate of 12\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>we have\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the height is increasing at the rate of\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> cm/s.\u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-14-sand-is-pouring-from-a-pipe-at-the-rate-of-12-cm-lt-sup-gt-3-lt-sup-gt-s-the-falling-sand-qna-11424850"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398490,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425560,"answerId":11425564,"questionText":"\u003cp>\u003cstrong>63. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> 3 \u003cstrong>(B)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cstrong>(C)\u003c/strong> -3 \u003cstrong>(D)\u003c/strong> - \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","answerText":"\u003cp>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Option (D) is correct.\u003c/p>","order":3,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-63-the-slope-of-the-normal-to-the-curve-y-2x2-3-sin-x-at-x-0-is-lt-strong-gt-lt-p-gt-lt-p-qna-11425560"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398562,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427308,"answerId":11427338,"questionText":"\u003cp>\u003cstrong>99. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cspan data-teams=\"true\">Kindly go through the solution\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734083769phptJNfSs.jpeg\" alt=\"\" width=\"574\" height=\"265\">\u003c/picture>\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734083797php9uWQPa.jpeg\" alt=\"\" width=\"505\" height=\"649\">\u003c/picture>\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734083815phpLheqCo.jpeg\" alt=\"\" width=\"244\" height=\"408\">\u003c/picture>\u003c/span>\u003c/p>","order":4,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-99-show-that-semi-vertical-angle-of-right-circular-cone-of-given-surface-area-and-maximum-qna-11427308"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398608,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427772,"answerId":11427776,"questionText":"\u003cp>\u003cstrong>122. The line y = mx + 1 is a tangent to the curve y\u003csup>2\u003c/sup> = 4x if the value of m is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> 1 \u003cstrong>(B)\u003c/strong> 2 \u003cstrong>(C)\u003c/strong> 3 \u003cstrong>(D)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","answerText":"\u003cp>The equation of the tangent to the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, substituting \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we get:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>m\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Since a tangent touches the curve at one point, the roots of equation (i) must be equal.\u003c/p>\u003cp>Therefore, we have:\u003c/p>\u003cp>Discriminant = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>m\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>m\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>m\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the required value of m is 1.\u003c/p>\u003cp>Therefore, option (A) is correct.\u003c/p>","order":5,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-122-the-line-y-mx-1-is-a-tangent-to-the-curve-y-lt-sup-gt-2-lt-sup-gt-amp-nbsp-4x-if-the-qna-11427772"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398406,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424952,"answerId":11424958,"questionText":"\u003cp>\u003cstrong>21. Show that the function given by f (x) = sin x is (a)\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> (b) decreasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(c) neither increasing nor decreasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = sin x.\u003c/p>\u003cp>So, f (x) = cosx.\u003c/p>\u003cp> (a) when, x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> i e, x in 1st quadrat.\u003c/p>\u003cp>f (x) = cos.x> 0\u003c/p>\u003cp>f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp> (b) when, x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> in IInd quadrat\u003c/p>\u003cp>f (x) = cosx< 0.\u003c/p>\u003cp>∴f (x) is strictly decreasing (π\u003c/p>\u003cp> (c) When, x \in (0, π).\u003c/p>\u003cp>f (x) = cosx is increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and decreasing\u003c/p>\u003cp>in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and f \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> = cos \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is neither increasing not decreasing in (0, π).\u003c/p>","order":6,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-21-show-that-the-function-given-by-f-x-sin-x-is-a-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-qna-11424952"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398428,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425130,"answerId":11425134,"questionText":"\u003cp>\u003cstrong>32. For what values of a the function f given by f(x) = x\u003csup>2\u003c/sup> + ax + 1 is increasing on [1, 2]?\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = x\u003csup>2\u003c/sup> + x + 1\u003c/p>\u003cp>So, f (x) = 2x + a\u003c/p>\u003cp>If f (x) is strictly increasing onx \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>, \u003c/mo>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, 1<x<2.< strong=\"\"></x<2.<>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo><\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmo><\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo><\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo><\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The minimum value of f (x) is 2 + a and that of men value is 4 + a.\u003c/p>\u003cp>∴ 2 +a> 0and 4 + a> 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> a> -2 and a> -4.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> a> -2.\u003c/p>\u003cp>∴The best value of a is -2.\u003c/p>","order":7,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-32-for-what-values-of-a-the-function-f-given-by-f-x-x-lt-sup-gt-2-lt-sup-gt-ax-1-is-increasing-qna-11425130"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398394,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424858,"answerId":11424862,"questionText":"\u003cp>\u003cstrong>15. The total cost C (x) in Rupees associated with the production of x units of an item is given by C (x) = 0.007x\u003csup>3\u003c/sup> - 0.003x\u003csup>2\u003c/sup> + 15x + 4000.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Find the marginal cost when 17 units are produced.\u003c/strong>\u003c/p>","answerText":"\u003cp>Given, c (x) = 0.007 \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 0.003\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 15x + 400.\u003c/p>\u003cp>Since the marginal cost is the rate of change of total cost wrt the output we have, \u003c/p>\u003cp>Marginal cost, MC, =\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dC\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 0.007 * 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 0.003 * 2\u003cem>x\u003c/em> + 15.\u003c/p>\u003cp>When x = 17, \u003c/p>\u003cp>Then, MC = 0.007 * 2. (17)\u003csup>2\u003c/sup> - 0.003 2 (17) + 15.\u003c/p>\u003cp>= 6.069 - 0.102 + 15.\u003c/p>\u003cp>= 20.967\u003c/p>\u003cp>Hence, the required marginal cost = ' 20, 97.\u003c/p>","order":8,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-15-the-total-cost-c-x-in-rupees-associated-with-the-production-of-x-units-of-an-item-is-given-qna-11424858"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398430,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425144,"answerId":11425148,"questionText":"\u003cp>\u003cstrong>33. Let I be any interval disjoint from [-1, 1]. Prove that the function f given by\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is increasing on I.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmover accent=\"true\">\u003cmi>x\u003c/mi>\u003cmo>? \u003c/mo>\u003c/mover>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, for every x\in I, where I is disjoint from [-1,1]\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>w\u003c/mi>\u003cmi>h\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = ( + ve) > 0 when x< -1.\u003c/p>\u003cp>∴f (x) is strictly increasing on I .\u003c/p>","order":9,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-33-let-i-be-any-interval-disjoint-from-prove-that-the-function-f-given-by-lt-strong-gt-lt-qna-11425144"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398452,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425326,"answerId":11425328,"questionText":"\u003cp>\u003cstrong>44. Find points at which the tangent to the curve y = x\u003csup>3\u003c/sup> - 3x\u003csup>2\u003c/sup> - 9x + 7 is parallel to the x-axis.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>slope of tangent to the given curve, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>when the tangent is parallel to x-axis \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cmath>\u003c/math> x = 3 or x = -1\u003c/p>\u003cp>When x = 3, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And when x = -1 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the required points are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":10,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-44-find-points-at-which-the-tangent-to-the-curve-y-x-lt-sup-gt-3-lt-sup-gt-ndash-3x-lt-sup-qna-11425326"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398408,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424968,"answerId":11424972,"questionText":"\u003cp>\u003cstrong>22. Find the intervals in which the function f given by f(x) = 2x\u003csup>2\u003c/sup> - 3x is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(a)\u003c/strong> Increasing \u003cstrong>(b) \u003c/strong>Decreasing\u003c/p>","answerText":"\u003cp>We have, f (x) = 2x\u003csup>2\u003c/sup> 3x\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Atf (x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 4x - 3 = 0\u003c/p>\u003cp>i e, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> divides the real line into two\u003c/p>\u003cp>disjoint interval \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong> (a) \u003c/strong>Now, \u003c/p>\u003cp>f (x) = 4x - 3 > 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong> (b) \u003c/strong>Now, f (x) = 4x - 3 < 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f (x) is strictly decreasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":11,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-22-find-the-intervals-in-which-the-function-f-given-by-f-x-2x-lt-sup-gt-2-lt-sup-gt-ndash-qna-11424968"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398410,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424986,"answerId":11424990,"questionText":"\u003cp>\u003cstrong>23. Find the intervals in which the function f given by f(x) = 2x\u003csup>3\u003c/sup> - 3x\u003csup>2\u003c/sup> - 36x + 7 is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(a) \u003c/strong>Increasing\u003cstrong> (b)\u003c/strong> Decreasing\u003c/p>","answerText":"\u003cp>We have, f (x) = 2\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 36\u003cem>x\u003c/em> + 7.\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 6 (\u003cem>x\u003c/em>\u003csup>2\u003c/sup>-\u003cem>x\u003c/em>- 6).\u003c/p>\u003cp>= 6 (\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3\u003cem>x\u003c/em> + 2\u003cem>x\u003c/em>- 6)\u003c/p>\u003cp>= 6 [x (\u003cem>x\u003c/em>- 3) + 2 (\u003cem>x\u003c/em>- 3)]\u003c/p>\u003cp>= 6 (\u003cem>x\u003c/em>- 3) (x + 2).\u003c/p>\u003cp>At, f (x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 6 (\u003cem>x\u003c/em>- 3) (x + 2) = 0.\u003c/p>\u003cp>So, when \u003cem>x\u003c/em>- 3 = 0 or x + 2 = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 3 or x = -2.\u003c/p>\u003cp>Hence we an divide the real line into three disjoint internal\u003c/p>\u003cp>I \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">) II\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)andIII\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) = ( + ve) ( -ve) ( -ve) = ( + ve) > 0.\u003c/p>\u003cp>So, f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, x\in ( -2,3), \u003c/p>\u003cp>f (x) = ( + ve) ( + ve) ( -ve) = ( -ve) < 0.\u003c/p>\u003cp>So, f (x) is strictly decreasing in ( -2,3).\u003c/p>\u003cp>At, x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) = ( + ve) ( + ve) ( + ve) = ( + )ve> 0.\u003c/p>\u003cp>So, f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴ (a) f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)and\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp> (b) f (x) is strictly decreasing in ( -2,3)\u003c/p>","order":12,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-23-find-the-intervals-in-which-the-function-f-given-by-f-x-2x-lt-sup-gt-3-lt-sup-gt-ndash-qna-11424986"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398402,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424922,"answerId":11424930,"questionText":"\u003cp>\u003cstrong>19. Show that the function given by f (x) = 3x + 17 is increasing on R.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, .\u003c/p>\u003cp>f (x) = 3x + 17.\u003c/p>\u003cp>So, f (x) = 3 > 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is strictly increasing on R.\u003c/p>","order":13,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-19-show-that-the-function-given-by-f-x-3x-17-is-increasing-on-r-lt-strong-gt-lt-p-gt-qna-11424922"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398404,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424940,"answerId":11424944,"questionText":"\u003cp>\u003cstrong>20. Show that the function given by f (x) = e2x is increasing on R.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = \u003cem>e\u003c/em>\u003csup>2\u003cem>x\u003c/em>\u003c/sup>\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> = 2\u003cem>e\u003c/em>\u003csup>2\u003cem>x\u003c/em>\u003c/sup>> 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is strictly increasing on R.\u003c/p>","order":14,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-20-show-that-the-function-given-by-f-x-e2x-is-increasing-on-r-lt-strong-gt-lt-p-gt-qna-11424940"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398414,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425022,"answerId":11425026,"questionText":"\u003cp>\u003cstrong>25. Show that 2 log(1 ) 2 x y x x = +- + , x > - 1, is an increasing function of x throughout its domain.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, y = log \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Differentiating the above wrt.x.we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The given domain of the given function isx> -1.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (x + 1) > 0.\u003c/p>\u003cp>Also, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(2 + \u003cem>x\u003c/em>)\u003csup>2\u003c/sup>> 0.\u003c/p>\u003cp>Hence, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=(+ve)\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴ y is an increasing function of x throughout its domain.\u003c/p>","order":15,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-25-show-that-2-log-1-2-x-y-x-x-minus-x-amp-gt-ndash-1-is-an-increasing-function-of-x-throughout-qna-11425022"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398412,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425010,"answerId":11425018,"questionText":"\u003cp>\u003cstrong>24. Find the intervals in which the following functions are strictly increasing or decreasing:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(a)\u003c/strong> x\u003csup>2\u003c/sup> + 2x - 5 \u003cstrong>(b)\u003c/strong> 10 - 6x - 2x\u003csup>2\u003c/sup>\u003cstrong> (c)\u003c/strong> -2x\u003csup>3\u003c/sup> - 9x\u003csup>2\u003c/sup> - 12x + 1\u003cstrong> (d)\u003c/strong> 6 - 9x - x\u003csup>2\u003c/sup>\u003c/p>\n\u003cp>\u003cstrong>(e)\u003c/strong> (x + 1)\u003csup>3\u003c/sup> (x - 3)\u003csup>3\u003c/sup> \u003c/p>","answerText":"\u003cp>\u003cstrong>(a)\u003c/strong> f (x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2x - 5.\u003c/p>\u003cp>f(x) = 2x + 2 = 2 (x + 1).\u003c/p>\u003cp>At, f(x) = 0\u003c/p>\u003cp>2 (x + 1) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = -1.\u003c/p>\u003cp>At, x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = (- ) ve< 0.\u003c/p>\u003cp>So, f (x)is strictly decreasing or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = ( + ve) > 1\u003c/p>\u003cp>f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(b)\u003c/strong> f(x) = 10 - 6\u003cem>x\u003c/em>- 2\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>So, f(x) = - 6 - 4\u003cem>x\u003c/em> = - 2 (3 + 2\u003cem>x\u003c/em>).\u003c/p>\u003cp>Atf(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 2 (3 + 2x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = ( -ve) ( + ve) = ( - ) ve< 0.\u003c/p>\u003cp>∴f (x) is strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(c) \u003c/strong>f (x) = 2\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 9\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em>.\u003c/p>\u003cp>So, f (x) =- 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 18\u003cem>x\u003c/em>- 12 = - 6 (\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 3\u003cem>x\u003c/em> + 2).\u003c/p>\u003cp>= -6 [\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>x\u003c/em> + 2\u003cem>x\u003c/em> + 2]\u003c/p>\u003cp>= -6 [x (x + 1) + 2 (x + 1)]\u003c/p>\u003cp>= -6 (x + 1) (x + 2)\u003c/p>\u003cp>At, f (x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 6 (x + 1) (x + 2) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = -1 and x = -2.\u003c/p>\u003cp>At, x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> , f(x) = ( -ve) ( -ve) ( -ve) = ( -ve) < 0\u003c/p>\u003cp>So, f(x) is strictly decreasing.\u003c/p>\u003cp>At, x( -2, -1), f(x) = ( -ve) ( -ve) ( -ve) = ( -ve) < 0.\u003c/p>\u003cp>So, f(x) is strictly decreasing\u003c/p>\u003cp>\u003cstrong>(d) \u003c/strong>f(x) = 6 - 9\u003cem>x\u003c/em>-\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>So, f(x) = - 9 - 2\u003cem>x\u003c/em>\u003c/p>\u003cp>At, f(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 9 - 2x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x =\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) > 0\u003c/p>\u003cp>So, f(x) is strictly increasing.\u003c/p>\u003cp>At x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) < 0.\u003c/p>\u003cp>So, f(x) is strictly decreasing.\u003c/p>\u003cp>\u003cstrong>(e) \u003c/strong>f(x) = (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em>- 3)\u003csup>3\u003c/sup>.\u003c/p>\u003cp>So, f(x) = (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup>. 3 (\u003cem>x\u003c/em>- 3)\u003csup>2\u003c/sup> + (\u003cem>x\u003c/em>- 3)\u003csup>3\u003c/sup>. 3 (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>\u003c/p>\u003cp>= 3 (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (\u003cem>x\u003c/em>- 3)\u003csup>2\u003c/sup> (\u003cem>x\u003c/em> + 1 + \u003cem>x\u003c/em>- 3).\u003c/p>\u003cp>= 3 (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (\u003cem>x\u003c/em>- 3)\u003csup>2\u003c/sup> (2\u003cem>x\u003c/em>- 2).\u003c/p>\u003cp>= 6 (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (\u003cem>x\u003c/em>- 3)\u003csup>2\u003c/sup> (\u003cem>x\u003c/em>- 1).\u003c/p>\u003cp>For strictly increasing,\u003c/p>\u003cp>f(x) > 0.\u003c/p>\u003cp>As 6 > 0,(\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>> 0 and (\u003cem>x\u003c/em>- 3)\u003csup>2\u003c/sup>> 0. We need\u003c/p>\u003cp>(x 1) > 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x> 1.\u003c/p>\u003cp>∴f (x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":16,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-24-find-the-intervals-in-which-the-following-functions-are-strictly-increasing-or-decreasing-qna-11425010"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398448,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425310,"answerId":11425314,"questionText":"\u003cp>\u003cstrong>42. Find the slope of the normal to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The Equation of the given curve are\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> which is the slope of the tanget to the curve.\u003c/p>\u003cp>Now, required slope of normal to the curve =\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003cbr>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>\u003c/mo>\u003cmn>Slopeoftangent\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cbr>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":17,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-42-find-the-slope-of-the-normal-to-the-curve-lt-span-title-quot-click-to-copy-mathml-quot-qna-11425310"},"anaFaqSource":"manual","generationSource":"manual","templateId":167,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398548,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427158,"answerId":11427164,"questionText":"\u003cp>\u003cstrong>92. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let A.B.C.D.be the square increased in a given fixed circle with radius x\u003c/p>\u003cp>Let ‘x’ and ‘y’ be the length and breadth of the rectangle\u003c/p>\u003cp>∴x, y> 0\u003c/p>\u003cp>In ABC, right angle at B, \u003c/p>\u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = (2\u003cem>x\u003c/em>)\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734082043php6kapSo.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734082028phpxXg4rZ.jpeg\" alt=\"\" width=\"283\" height=\"254\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734082043php6kapSo_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734082043php6kapSo.jpeg\" alt=\"\" width=\"362\" height=\"122\">\u003c/picture>\u003c/div>\u003c/div>","order":18,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-92-show-that-of-all-the-rectangles-inscribed-in-a-given-fixed-circle-the-square-has-the-maximum-qna-11427158"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398424,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425082,"answerId":11425092,"questionText":"\u003cp>\u003cstrong>30. Which of the following functions are decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> ?\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A) \u003c/strong>cos x \u003cstrong>(B) \u003c/strong>cos 2x \u003cstrong>(C)\u003c/strong> cos 3x \u003cstrong>(D) \u003c/strong>tan x\u003c/p>","answerText":"\u003cp>\u003cstrong>(A) \u003c/strong>We have,\u003c/p>\u003cp>f(x) = cosx\u003c/p>\u003cp>So, f(x) = -sin x\u003c/p>\u003cp>When x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> we know that sin x> 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -sinx< 0. f(x) < 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f(x) is strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(B)\u003c/strong> We have, f(x) = cos 2x\u003c/p>\u003cp>So, f(x) = -2sin 2x\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> we know that sin x> 0.\u003c/p>\u003cp>i e, 0 <x< <span=\"\" class=\"mathml\" title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003cbr>=> 0<2x<π </x<>\u003c/p>\u003cp>So, sin 2x> 0 (sinØ is ( +ve) in 1st and 2ndquadrant).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -2sin 2x< 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) < 0.\u003c/p>\u003cp>∴f (x) is strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(c) \u003c/strong>We have, f(x) = cos 3x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) = -3 sin 3x.\u003c/p>\u003cp>As \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>We can divide the interval into two\u003c/p>\u003cp>Case I At, 0 < 3x\u003c/p>\u003cp>sin 3x> 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -3 sin 3x< 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmi>B\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmo>\to\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) < 0.\u003c/p>\u003cp>∴f(x) is strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>'\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>case II. At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>w\u003c/mi>\u003cmi>e\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>g\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>t\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>i\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmi>d\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>q\u003c/mi>\u003cmi>u\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>u\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>sin 3x< 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -3 sin 3x> 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) > 0. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo><\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, f(x) is with a increasing or decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(d) \u003c/strong>We have, f(x) = tan x.\u003c/p>\u003cp>So, f(x) = sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) = sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>> 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\forall\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the \u003cem>fx\u003c/em>\u003csup>n\u003c/sup>cosx and cos 2x are strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":19,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-30-which-of-the-following-functions-are-decreasing-on-lt-span-title-quot-click-to-copy-mathml-qna-11425082"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398558,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427262,"answerId":11427268,"questionText":"\u003cp>\u003cstrong>97. Show that the right circular cone of least curved surface and given volume \u003c/strong>\u003cstrong>has an altitude equal to \sqrt2\u003c/strong>\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>\u003cstrong> time the radius of the base.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r and h be the radius and height of the cone.\u003c/p>\u003cp>The volume V of the cone is.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3V\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>k (SAY)\u003c/mi>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And curve surface area S is\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734082893php6RewPt.jpeg\" alt=\"\" width=\"523\" height=\"634\">\u003c/picture>\u003c/div>\u003c/div>","order":20,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-97-show-that-the-right-circular-cone-of-least-curved-surface-and-given-volume-lt-strong-gt-qna-11427262"},"anaFaqSource":"manual","generationSource":"manual","templateId":141,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398366,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424714,"answerId":11424718,"questionText":"\u003cp>\u003cstrong>1. Find the rate of change of the area of a circle with respect to its radius r when\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>) = 2πr.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong> (a) \u003c/strong>r = 3cm\u003c/p>\u003cp>When r = 3 cm, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 2 \times π \times 3 cm = 6π.\u003c/p>\u003cp>Thus, the area of the circle is changing at the rate of 6π cm \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>.\u003c/p>\u003cp> (b) r = 4cm\u003c/p>\u003cp>whenr = 4cm, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 2 \times π \times 4cm = 8π.\u003c/p>\u003cp>Thus, the area of the circle is changing at the rate of 8 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>.\u003c/p>","order":21,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-1-find-the-rate-of-change-of-the-area-of-a-circle-with-respect-to-its-radius-r-when-lt-strong-qna-11424714"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398426,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425106,"answerId":11425120,"questionText":"\u003cp>\u003cstrong>31. On which of the following intervals is the function f given by f(x) = x\u003csup>100\u003c/sup> + sin x -1 decreasing ?\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> (0,1) \u003cstrong>(B)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> \u003cstrong>(C)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> \u003cstrong>(D)\u003c/strong> None of these\u003c/p>","answerText":"\u003cp>We have, f (x) = x 100 + sin x - 1.\u003c/p>\u003cp>So, f (x) = 100\u003cem>x\u003c/em>\u003csup>99\u003c/sup> + cosx.\u003c/p>\u003cp>\u003cstrong>(A) \u003c/strong>When x (0,1). We get.\u003c/p>\u003cp>x>0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 100 \u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 0.\u003c/p>\u003cp>Now, 0 radian = 0 degree\u003c/p>\u003cp>and 1 radian = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>°\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo>\times\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>°\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>°\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, cosx> 0 for x\in (0,1) radian = (0,57)\u003c/p>\u003cp>∴f (x) > 0 for x\in (0,1).\u003c/p>\u003cp>\u003cstrong>(B) \u003c/strong>When x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> we get,\u003c/p>\u003cp>So, x> 1\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 1\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 100\u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 100. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>?\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\"> which is great than \u003cmn>1\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And cosx is negative between -1 and 0.\u003c/p>\u003cp>So, f (x) = 100\u003cem>x\u003c/em>\u003csup>99\u003c/sup> + cos\u003cem>x\u003c/em>> 100 - 1 = 99 > 0.\u003c/p>\u003cp>∴f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(c)\u003c/strong> When x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x> 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 100\u003cem>x\u003c/em>\u003csup>99\u003c/sup>> 0\u003c/p>\u003cp>and cosx> 0. (firstquadrant).\u003c/p>\u003cp>I e, f(x) > 0.\u003c/p>\u003cp>∴f(x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, option (D) is correct.\u003c/p>","order":22,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-31-on-which-of-the-following-intervals-is-the-function-f-given-by-f-x-x-lt-sup-gt-100-lt-qna-11425106"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398440,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425258,"answerId":11425262,"questionText":"\u003cp>\u003cstrong>Q38. Find the slope of the tangent to the curve y = 3x\u003csup>4\u003c/sup> - 4x at x = 4.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of the tangent at x = 4 is given by\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>6\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 764\u003c/p>","order":23,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-q38-find-the-slope-of-the-tangent-to-the-curve-y-3x-lt-sup-gt-4-lt-sup-gt-ndash-4x-at-x-4-qna-11425258"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398488,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425550,"answerId":11425556,"questionText":"\u003cp>\u003cstrong>62. Find the equation of the tangent to the curve y = \sqrt3x - 2 which is parallel to the line 4x - 2y + 5 = 0.\u003c/strong>\u003c/p>","answerText":"\u003cdiv class=\"photo-widget-full\">\u003cdiv class=\"figure\">\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003c/picture>\u003c/div>\u003cdiv class=\"figure\">\u003cspan data-teams=\"true\">Kindly go through the solution\u003c/span>\u003c/div>\u003cdiv class=\"figure\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/1748933077phpE7OjTK.jpeg\" alt=\"\" width=\"748\" height=\"388\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1733995822phpqJVfEv.jpeg\" alt=\"\" width=\"393\" height=\"640\">\u003c/picture>\u003c/p>","order":24,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-62-find-the-equation-of-the-tangent-to-the-curve-y-radic-3x-2-which-is-parallel-to-the-line-qna-11425550"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398400,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424896,"answerId":11424902,"questionText":"\u003cp>\u003cstrong>18. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A) \u003c/strong>116\u003cstrong> (B)\u003c/strong> 96 \u003cstrong>(C) \u003c/strong>90\u003cstrong> (D)\u003c/strong> 126\u003c/p>","answerText":"\u003cp>Given, R (x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 36x + 5.\u003c/p>\u003cp>Marginal revenue, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>R\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 3 * 2x + 36\u003c/p>\u003cp>= 6x + 36\u003c/p>\u003cp>When x = 15.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">R (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>*\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> = 90 + 36 = 126\u003c/p>\u003cp>Option (D) is correct.\u003c/p>","order":25,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-18-the-total-revenue-in-rupees-received-from-the-sale-of-x-units-of-a-product-is-given-by-qna-11424896"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398422,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425064,"answerId":11425074,"questionText":"\u003cp>\u003cstrong>29. Prove that the function f given by f(x) = x\u003csup>2\u003c/sup> - x + 1 is neither strictly increasing nor decreasing on (- 1, 1).\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup>-\u003cem>x\u003c/em> + 1.\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Atf (x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 2x - 1 = 0\u003c/p>\u003cp>I e, x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> divides the real line into two disjoint interval the interval ( -1, 1) into\u003c/p>\u003cp>Two disjoint interval\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And f (x) is strictly increasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and strictly decreasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, f (x) is with a increasing or decreasing on ( -1,1).\u003c/p>","order":26,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-29-prove-that-the-function-f-given-by-f-x-x-lt-sup-gt-2-lt-sup-gt-ndash-x-1-is-neither-strictly-qna-11425064"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398386,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424822,"answerId":11424826,"questionText":"\u003cp>\u003cstrong>11. A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate\u003c/strong>\u003c/p>","answerText":"\u003cp>Given eq\u003csup>n\u003c/sup> of the curve is 6y = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 2.______ (1)\u003c/p>\u003cp>Wheny coordinate change s 8 times as fast as x-coordinate\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 8 _____ (2)\u003c/p>\u003cp>Now, differentiating eq\u003csup>n \u003c/sup> (1) wrt.x we get, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>6\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>6 \times 8 = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> (using eq\u003csup>n\u003c/sup> (2)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>+\sqrtx\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>=\u003c/mo>\u003cmo>\pm\sqrt16\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\pm\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>4.\u003c/p>\u003cp>When x = 4, we have, 6y = 4\u003csup>3\u003c/sup>+ 2 = 64 + 2 + 66\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> y =11.\u003c/p>\u003cp>And when x = -4, we have, 6y = ( -4)3 + 2 = -64 + 2 = -62\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The tequired point s are (4, 11) and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":27,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-11-a-particle-moves-along-the-curve-6y-x3-2-find-the-points-on-the-curve-at-which-the-y-coordinate-qna-11424822"},"anaFaqSource":"manual","generationSource":"manual","templateId":129,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398390,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424840,"answerId":11424844,"questionText":"\u003cp>\u003cstrong>13. A balloon, which always remains spherical, has a variable diameter 3 (2 1) 2 x + . Find the rate of change of its volume with respect to x.\u003c/strong>\u003c/p>","answerText":"\u003cp>Given, diameter of the spherical balloon = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (2x + 1)\u003c/p>\u003cp>So, radius of the spherical r = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Then, volume of the spherical V = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Q Rate of change of volume wrt.tox, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>V\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\cdot\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\times\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":28,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-13-a-balloon-which-always-remains-spherical-has-a-variable-diameter-3-2-1-2-x-find-the-rate-qna-11424840"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398416,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425032,"answerId":11425038,"questionText":"\u003cp>\u003cstrong>26. Find the values of x for which y = [x(x - 2)]2 is an increasing function.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, y = [\u003cem>x\u003c/em> (\u003cem>x\u003c/em>- 2)]\u003csup>2\u003c/sup>.\u003c/p>\u003cp>Differentiating the above w rt. x we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2 [\u003cem>x\u003c/em> (\u003cem>x\u003c/em>- 2)] (\u003cem>x\u003c/em> + \u003cem>x\u003c/em>- 2)\u003c/p>\u003cp>= 2x (x - 2) (2x - 2)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 4x (x - 2) (x - 1).\u003c/p>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 4x (x - 2) (x - 1) = 0.\u003c/p>\u003cp>i e, x = 0, x = 2, x = 1 divides the real line into\u003c/p>\u003cp>four disjoint interval. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> [0,1],[1,2] and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>when x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\leq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is decreasing in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is increasing in [0,1].\u003c/p>\u003cp>When x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\leq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>0for \u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is decreasing.\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\geq\u003c/mo>\u003cmn>0,\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>,for \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is in creasing\u003c/p>\u003cp>Hence, f (x) is increasing for x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":29,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-26-find-the-values-of-x-for-which-y-2-is-an-increasing-function-lt-strong-gt-lt-p-gt-qna-11425032"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398420,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425056,"answerId":11425058,"questionText":"\u003cp>\u003cstrong>28. Prove that the logarithmic function is increasing on (0, \infty).\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = log x.\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>i e, f (x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> > 0. When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the logarithmic fx is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":30,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-28-prove-that-the-logarithmic-function-is-increasing-on-0-infin-lt-strong-gt-lt-p-gt-qna-11425056"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398432,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425156,"answerId":11425164,"questionText":"\u003cp>\u003cstrong>34. Prove that the function f given by f(x) = log sin x is increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = log sin x\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) = cot x> 0 (Ist quadrat )\u003c/p>\u003cp>So, f (x) is strictly increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) = cot x< 0. (IIndquadrant ).\u003c/p>\u003cp>So, f (x) is strictly decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":31,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-34-prove-that-the-function-f-given-by-f-x-log-sin-x-is-increasing-on-lt-span-title-quot-click-qna-11425156"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398434,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425172,"answerId":11425180,"questionText":"\u003cp>\u003cstrong>35. Prove that the function f given by f (x) = log |cos x| is decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = log \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>whenx \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we get.\u003c/p>\u003cp>tanx> 0 (Ist quadrant).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> tanx< 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) < 0.\u003c/p>\u003cp>∴f (x) is decreasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> we get, \u003c/p>\u003cp>tanx|< |0 (ivth quadrant).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -tanx|>| 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) > 0\u003c/p>\u003cp>∴f (x) is increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":32,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-35-prove-that-the-function-f-given-by-f-x-log-cos-x-is-decreasing-on-lt-span-title-quot-click-qna-11425172"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398438,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425204,"answerId":11425208,"questionText":"\u003cp>\u003cstrong>37. The interval in which y = x\u003csup>2\u003c/sup> e\u003csup>-x\u003c/sup> is increasing is (A) (- \infty, \infty) (B) (- 2, 0) (C) (2, \infty) (D) (0, 2)\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = x\u003csup>2\u003c/sup> e\u003csup>-x\u003c/sup>\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmn>2\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmi>x\u003c/mi>\u003cmenclose notation=\"updiagonalstrike\">\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/menclose>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= -\u003cem>x\u003c/em>\u003csup>2\u003c/sup> e\u003csup>-\u003cem>x\u003c/em>\u003c/sup> + e\u003csup>-\u003cem>x\u003c/em>\u003c/sup> 2\u003cem>x\u003c/em>.\u003c/p>\u003cp>= \u003cem>x\u003c/em> e\u003csup>-\u003cem>x\u003c/em>\u003c/sup> ( x + 2).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>e\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>If f (x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 0, x = 2.\u003c/p>\u003cp>Hence, we get there disjoint interval\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> [\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>, \u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When, x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo>, \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we have, f (x) = ( -ve) ( + ve) = ( -ve) < 0.\u003c/p>\u003cp>So, f is strictly decreasing.\u003c/p>\u003cp>When x \in (0,2), f (x) = ( + ve) ( + ve) = ( + ve) > 0.\u003c/p>\u003cp>So, f is strictly increasing.\u003c/p>\u003cp>And when x \in\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmo>\infty\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) = ( +ve) ( -ve) = ( -ve) < 0.\u003c/p>\u003cp>So, f is strictly decreasing.\u003c/p>\u003cp>Hence, option (D) is correct.\u003c/p>","order":33,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-37-the-interval-in-which-y-x-lt-sup-gt-2-lt-sup-gt-e-lt-sup-gt-ndash-x-lt-sup-gt-is-increasing-qna-11425204"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398496,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425618,"answerId":11425622,"questionText":"\u003cp>\u003cstrong>66. Find the approximate value of f(2.01), where f (x) = 4x\u003csup>2\u003c/sup> + 5x + 2.\u003c/strong>\u003c/p>","answerText":"\u003cp>Given, y = f (x) = 4x\u003csup>2\u003c/sup> + 5x + 2.\u003c/p>\u003cp>So, f (x) = 8x + 5. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 8x + 5 dy = (8x + 5) dx.\u003c/p>\u003cp>Let x = 2 and Δx = 0.01.Then, \u003c/p>\u003cp>f (x + Δx) = f (2 + 0.01) = f (2.01).\u003c/p>\u003cp>Δy = f. (x + Δx) f (Δx).\u003c/p>\u003cp>f (x +Δx) = f (x) +Δy.\u003c/p>\u003cp>= f (x) + dy = f (x) + (8x + 5) dx.\u003c/p>\u003cp>= f (2.01) = f (2) + (8 x 2 + 5). Δx {∴dx = Δx}\u003c/p>\u003cp>= 4 (2)\u003csup>2\u003c/sup> + 5 (2) + 2 + 21 (0.01)\u003c/p>\u003cp>= 16 + 10 + 2 + 0.21 = 28.21.\u003c/p>","order":34,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-66-find-the-approximate-value-of-f-2-01-where-f-x-4x-lt-sup-gt-2-lt-sup-gt-5x-2-lt-strong-qna-11425618"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398500,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425634,"answerId":11425636,"questionText":"\u003cp>\u003cstrong>68. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.\u003c/strong>\u003c/p>","answerText":"\u003cp>We know that, the volume v of side ‘a’ mete of cube is v = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>.\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\cdot\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\cdot\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given that, increase in side = 1% of x.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo> m\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":35,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-68-find-the-approximate-change-in-the-volume-v-of-a-cube-of-side-x-metres-caused-by-increasing-qna-11425634"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398528,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426954,"answerId":11426968,"questionText":"\u003cp>\u003cstrong>82. What is the maximum value of the function sin x + cos x?\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = sin x + cos x.\u003c/p>\u003cp>f (x) = cos x - sin x.\u003c/p>\u003cp>At f (x) = 0\u003c/p>\u003cp>cosx - sin x = 0\u003c/p>\u003cp>sinx = cos x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> , \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734073563phpXVSB7z_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734073563phpXVSB7z.jpeg\" alt=\"\" width=\"262\" height=\"107\">\u003c/picture>\u003c/div>\u003c/div>","order":36,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-82-what-is-the-maximum-value-of-the-function-sin-x-cos-x-lt-strong-gt-lt-p-gt-qna-11426954"},"anaFaqSource":"manual","generationSource":"manual","templateId":91,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398534,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427008,"answerId":11427014,"questionText":"\u003cp>\u003cstrong>85. Find the maximum and minimum values of x + sin 2x on [0, 2π].\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f(x) = x + sin 2x ,x \in [0, 2π].\u003c/p>\u003cp>f(x) = 1 + 2cos 2x\u003c/p>\u003cp>At f(x) = 0\u003c/p>\u003cp>1 + 2 cos2x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>n\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>d\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> 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stretchy=\"false\">]\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>d\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> 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mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\sqrt3/2\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 1.05 + 0.87\u003c/p>\u003cp>= 1.92\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 1.23\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=5.07.\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 5.25 - 0.87 = 4.38\u003c/p>\u003cp>At and points,\u003c/p>\u003cp>f(0) = 0 + sin2 \times 0 = 0\u003c/p>\u003cp>f(2π) = 2π + sin 2 \times 2π = 6.2 + 0 = 6.28\u003c/p>\u003cp>∴Maximum value of f(x) = 6.28 at x = 2π and\u003c/p>\u003cp>minimum value of f(x) = 0 at x= 0\u003c/p>","order":37,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-85-find-the-maximum-and-minimum-values-of-x-sin-2x-on-lt-strong-gt-lt-p-gt-qna-11427008"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398604,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427734,"answerId":11427738,"questionText":"\u003cp>\u003cstrong>120. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>α\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cspan data-teams=\"true\">Kindly go through the solution\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734093179php9W4ar2.jpeg\" alt=\"\" width=\"480\" height=\"454\">\u003c/picture>\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734093194phpKMnn8r.jpeg\" alt=\"\" width=\"261\" height=\"97\">\u003c/picture>\u003c/span>\u003c/p>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734093208phpNsNZiV.jpeg\" alt=\"\" width=\"348\" height=\"442\">\u003c/picture>\u003c/span>\u003c/p>","order":38,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-120-show-that-height-of-the-cylinder-of-greatest-volume-which-can-be-inscribed-in-a-right-qna-11427734"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398610,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427786,"answerId":11427792,"questionText":"\u003cp>\u003cstrong>123. The normal at the point (1,1) on the curve 2y + x\u003csup>2\u003c/sup> = 3 is (A) x + y = 0 (B) x - y = 0 (C) x + y +1 = 0 (D) x - y = 1\u003c/strong>\u003c/p>","answerText":"\u003cp>The equation of the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>Differentiate with respect to x, we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>The slope of the normal to the given curve at point (1,1) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the equation of the normal to the given curve at (1,1) is given as:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Therefore, option (B) is correct.\u003c/p>","order":39,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-123-the-normal-at-the-point-1-1-on-the-curve-2y-x-lt-sup-gt-2-lt-sup-gt-3-is-a-x-y-0-b-x-qna-11427786"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398368,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424728,"answerId":11424734,"questionText":"\u003cp>\u003cstrong>2. The volume of a cube is increasing at the rate of 8 cm3 /s. How fast is the surface area increasing when the length of an edge is 12 cm?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x be the length of edge,v be the value and s be the surface area of the cube then,\u003c/p>\u003cp>y = x3.\u003c/p>\u003cp>and S = 6x2, where x is a fxn of time.\u003c/p>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>Y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>F\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (x3) = 8\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>\cdot\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> (by chain rule)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 3x2 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (bx2) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 12x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>.\u003c/p>\u003cp>When x = 12 cm,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>.\u003c/p>","order":40,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-2-the-volume-of-a-cube-is-increasing-at-the-rate-of-8-cm3-s-how-fast-is-the-surface-area-qna-11424728"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398370,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424744,"answerId":11424746,"questionText":"\u003cp>\u003cstrong>3. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘r’ cm be the radius of the circle which is afxn of time.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 8 3cm/s as it is increasing.\u003c/p>\u003cp>Now, the area A of the circle is A = πr2.\u003c/p>\u003cp>So, the rate at which the area of the circle change \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> πr2.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmtext> \u003c/mtext>\u003cmo>\cdot\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2πr 3\u003c/p>\u003cp>= 6πr. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When r = 10cm, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 6.π \times 10 = 60π \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":41,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-3-the-radius-of-a-circle-is-increasing-uniformly-at-the-rate-of-3-cm-s-find-the-rate-at-which-qna-11424744"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398372,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424754,"answerId":11424760,"questionText":"\u003cp>\u003cstrong>4. An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let 'x' cm be the length of edge of the cube which is a \u003cem>fx\u003csup>n\u003c/sup>\u003c/em> of time t then, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 3cm/s as it is increasing.\u003c/p>\u003cp>Now, volume v of the cube is v = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>\u003c/p>\u003cp>Ø Rate of change of volume of the cube \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>q\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.3\u003c/p>\u003cp>= 9\u003cem>x\u003c/em>\u003csup>2\u003c/sup> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When x = 10cm.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 9 x (10)\u003csup>2\u003c/sup>= 900 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":42,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-4-an-edge-of-a-variable-cube-is-increasing-at-the-rate-of-3-cm-s-how-fast-is-the-volume-of-qna-11424754"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398374,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424766,"answerId":11424772,"questionText":"\u003cp>\u003cstrong>5. A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?\u003c/strong>\u003c/p>","answerText":"\u003cp>The area A of the circle with radius π is A = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>Then, rate of change in area \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmover accent=\"true\">\u003cmi>x\u003c/mi>\u003cmo>? \u003c/mo>\u003c/mover>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>\cdot\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2πr \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmi>dr\u003c/mi>\u003c/math>\u003cbr>\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Q The wave moves at a rate 5cm/s we have, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>dF\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 5cm/s\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> =r. 5 = 10πr. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When r = 8 cm.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 10.π.8 cm \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = 80 \times cm \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":43,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-5-a-stone-is-dropped-into-a-quiet-lake-and-waves-move-in-circles-at-the-speed-of-5-cm-s-at-qna-11424766"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398376,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424778,"answerId":11424780,"questionText":"\u003cp>\u003cstrong>6. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?\u003c/strong>\u003c/p>","answerText":"\u003cp>The circumference C of the circle with radius r is C = 2πr.\u003c/p>\u003cp>Then, rate of change in circumference is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dC\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>f\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Q Radius of circle increases at rate 0.7 cm/s we get, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> cm/s\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dC\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 2.\times 0.7 cm/s = 1.4 \times cm/s.\u003c/p>","order":44,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-6-the-radius-of-a-circle-is-increasing-at-the-rate-of-0-7-cm-s-what-is-the-rate-of-increase-qna-11424778"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398378,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424784,"answerId":11424792,"questionText":"\u003cp>\u003cstrong>7. The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.\u003c/strong>\u003c/p>","answerText":"\u003cp>Since the length x is decreasing and the widthy is increasing with respect to time we have\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 5cm/min and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = A cm/min\u003c/p>\u003cp> (a) The perimeter P of a rectangle will be, P = 2 (x + y)\u003c/p>\u003cp>Q Rate of change of perimeter, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>P\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2 ( -5 + 4)\u003c/p>\u003cp>= -2 cm/min\u003c/p>\u003cp> (b) The area A of the rectangle is A = x. y.\u003c/p>\u003cp>Rate of change of area is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>\cdot\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 4x - 5y\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> |x = 8ay = 6cm = 4 (8) - 5 (6) = 32 - 30 = 2 cm\u003csup>2\u003c/sup>/min spherical balloon\u003c/p>","order":45,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-7-the-length-x-of-a-rectangle-is-decreasing-at-the-rate-of-5-cm-minute-and-the-width-y-is-qna-11424784"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398380,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424798,"answerId":11424800,"questionText":"\u003cp>\u003cstrong>8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘r’ cm be the radius of volume V. measured Then,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>V =\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, rate at which balloon is being inflated = 900\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> = 900 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> \times 3 \times \u003cem>r\u003c/em>\u003csup>2\u003c/sup> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 900.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When r = 15cm,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> cm/s.\u003c/p>\u003cp>Q Radius of balloon increases by \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> per second.\u003c/p>","order":46,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-8-a-balloon-which-always-remains-spherical-on-inflation-is-being-inflated-by-pumping-in-900-qna-11424798"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398382,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424806,"answerId":11424808,"questionText":"\u003cp>\u003cstrong>9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm\u003c/strong>\u003c/p>","answerText":"\u003cp>The volume v of a spherical balloon with radius r is V.\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>with respect its radius.\u003c/p>\u003cp>Then, the rate of change of volume \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\cdot\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 4\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>Whenx = 10 cm, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 4 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>10)\u003csup>2\u003c/sup> = 400 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>cm\u003csup>3\u003c/sup>/cm\u003c/p>","order":47,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-9-a-balloon-which-always-remains-spherical-has-a-variable-radius-find-the-rate-at-which-its-qna-11424806"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398384,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424812,"answerId":11424814,"questionText":"\u003cp>\u003cstrong>10. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?\u003c/strong>\u003c/p>","answerText":"\u003cp>Since, the bottom of ground is increasing with time t,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 2cm/s\u003c/p>\u003cp>From fig, Δ ABC, by Pythagorastheorem\u003c/p>\u003cp>AB\u003csup>2\u003c/sup> + BC\u003csup>2\u003c/sup> = AC\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 5\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 25 ____ (1)\u003c/p>\u003cp>Differentiating eq\u003csup>n\u003c/sup> (1) w. r. t. time t we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> m/s\u003c/p>\u003cp>When x = 4m, the rate at which its height on the wall decreases is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \sqrt9\u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmi>L\u003c/mi>\u003cmtext> engthcan'tbenegetive\u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> room\u003c/p>","order":48,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-10-a-ladder-5-m-long-is-leaning-against-a-wall-the-bottom-of-the-ladder-is-pulled-along-the-qna-11424812"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398388,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424834,"answerId":11424836,"questionText":"\u003cp>\u003cstrong>12. The radius of an air bubble is increasing at the rate of 1 2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x be the radius of the bubble with volume .V. then, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> cm/s\u003c/p>\u003cp>andV = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Rate of change of volume \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>V\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 4π\u003cem>r\u003c/em>\u003csup>2\u003c/sup> \times\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>=\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span> = 2\u003cem>x\u003c/em> (1)\u003csup>2\u003c/sup> 2π. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>cm\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":49,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-12-the-radius-of-an-air-bubble-is-increasing-at-the-rate-of-1-2-cm-s-at-what-rate-is-the-qna-11424834"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398396,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424874,"answerId":11424878,"questionText":"\u003cp>\u003cstrong>16. The total revenue in Rupees received from the sale of x units of a product is given by R (x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.\u003c/strong>\u003c/p>","answerText":"\u003cp>Given, R (x) = 13\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 26\u003cem>x\u003c/em> + 15.\u003c/p>\u003cp>Marginal revenue is the rate of change of total revenue with respect to the number of units sold Marginal revenue (MR) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>R\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 13 * 2x + 26\u003c/p>\u003cp>= 26x + 26\u003c/p>\u003cp>When x = 7, \u003c/p>\u003cp>MR = 26 * 7 + 26 = 182 + 26 = 208.\u003c/p>\u003cp>Hence, the required marginal reverse = ' 208.\u003c/p>\u003cp>Choose the correct answer for questions 17 and 18.\u003c/p>","order":50,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-16-the-total-revenue-in-rupees-received-from-the-sale-of-x-units-of-a-product-is-given-by-qna-11424874"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398398,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11424882,"answerId":11424886,"questionText":"\u003cp>\u003cstrong>17. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> 10π \u003cstrong>(B) \u003c/strong>12π \u003cstrong>(C) \u003c/strong>8π \u003cstrong>(D)\u003c/strong> 11π\u003c/p>","answerText":"\u003cp>The area A of the isle with radius r is given by with respect to radius r A = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>Then, rate of change of area of the circle \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmo>\cdotA\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2πr.\u003c/p>\u003cp>When r = 6 cm\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Q option (B) is correct.\u003c/p>","order":51,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-17-the-rate-of-change-of-the-area-of-a-circle-with-respect-to-its-radius-r-at-r-6-cm-is-lt-qna-11424882"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398442,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425272,"answerId":11425274,"questionText":"\u003cp>\u003cstrong>39. Find the slope of the tangent to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent at x = 10 is given by,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":52,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-39-find-the-slope-of-the-tangent-to-the-curve-lt-span-title-quot-click-to-copy-mathml-quot-qna-11425272"},"anaFaqSource":"manual","generationSource":"manual","templateId":167,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398444,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425284,"answerId":11425288,"questionText":"\u003cp>\u003cstrong>40. Find the slope of the tangent to curve y = x\u003csup>3\u003c/sup> - x + 1 at the point whose x- coordinate is 2.\u003c/strong>\u003c/p>","answerText":"\u003cp>Slope of tangent to the given curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":53,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-40-find-the-slope-of-the-tangent-to-curve-y-x-lt-sup-gt-3-lt-sup-gt-ndash-x-1-at-the-point-qna-11425284"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398446,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425296,"answerId":11425304,"questionText":"\u003cp>\u003cstrong>41. Find the slope of the tangent to the curve y = x\u003csup>3\u003c/sup> -3x + 2 at the point whose x-coordinate is 3.\u003c/strong>\u003c/p>","answerText":"\u003cp>Slope of tangent to the given curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>so, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":54,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-41-find-the-slope-of-the-tangent-to-the-curve-y-x-lt-sup-gt-3-lt-sup-gt-ndash-3x-2-at-the-qna-11425296"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398450,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425320,"answerId":11425322,"questionText":"\u003cp>\u003cstrong>43. Find the slope of the normal to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curves are\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>so, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent to curve at \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, slope of normal to curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":55,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-43-find-the-slope-of-the-normal-to-the-curve-lt-span-title-quot-click-to-copy-mathml-quot-qna-11425320"},"anaFaqSource":"manual","generationSource":"manual","templateId":167,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398454,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425336,"answerId":11425342,"questionText":"\u003cp>\u003cstrong>45. Find a point on the curve y = (x - 2)\u003csup>2\u003c/sup> at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4)\u003c/strong>\u003c/p>","answerText":"\u003cp>Let the point joining the chord be \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Then slope of the chord \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo> {\u003c/mo>\u003cmrow>\u003cmo>? \u003c/mo>\u003cmo>Slope=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2\u003c/p>\u003cp>The given eq\u003csup>n\u003c/sup> of the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> \u003c/p>\u003cp>slope of the tangent to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given that, the tangent is parallel to the chord PQ.\u003c/p>\u003cp>slope of tangent = slope of PQ.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The required point on curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":56,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-45-find-a-point-on-the-curve-y-x-ndash-2-lt-sup-gt-2-lt-sup-gt-at-which-the-tangent-is-parallel-qna-11425336"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398456,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425348,"answerId":11425352,"questionText":"\u003cp>\u003cstrong>46. Find the point on the curve y = x\u003csup>3\u003c/sup> - 11x + 5 at which the tangent is y = x - 11.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>slope of tangent to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Then eq\u003csup>n\u003c/sup> of tangent is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> which gives us slope \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When x = 2, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And when x = 2, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> when put into \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we get\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> which is true.\u003c/p>\u003cp>and the point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> when put into \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> gives,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> which is not true.\u003c/p>\u003cp>Hence, the required point is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003c/p>","order":57,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-46-find-the-point-on-the-curve-y-x-lt-sup-gt-3-lt-sup-gt-ndash-11x-5-at-which-the-tangent-qna-11425348"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398458,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425360,"answerId":11425366,"questionText":"\u003cp>\u003cstrong>47. Find the equation of all lines having slope - 1 that are tangents to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent to the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given that, slope of tangent = 1.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>\pm\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>ie, X=1+1\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and when \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the point of contact of the tangents are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)and\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The reqd. eq\u003csup>n\u003c/sup> of line are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>?\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003cmtd columnalign=\"center\"> eqn\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup> of line \u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":58,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-47-find-the-equation-of-all-lines-having-slope-ndash-1-that-are-tangents-to-the-curve-lt-qna-11425360"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398460,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425374,"answerId":11425378,"questionText":"\u003cp>\u003cstrong>48. Find the equation of all lines having slope 2 which are tangents to the curve\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent to the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> which is not possible\u003c/p>\u003cp>we conclude that there is no possible tangent to the given curve with slope = 2.\u003c/p>","order":59,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-48-find-the-equation-of-all-lines-having-slope-2-which-are-tangents-to-the-curve-lt-strong-qna-11425374"},"anaFaqSource":"manual","generationSource":"manual","templateId":119,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398462,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425390,"answerId":11425398,"questionText":"\u003cp>\u003cstrong>49. Find the equations of all lines having slope 0 which are tangent to the curve\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent to the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The point of contact of the tangent to the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The eq\u003csup>n\u003c/sup> of the line is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":60,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-49-find-the-equations-of-all-lines-having-slope-0-which-are-tangent-to-the-curve-lt-strong-qna-11425390"},"anaFaqSource":"manual","generationSource":"manual","templateId":95,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398464,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425402,"answerId":11425408,"questionText":"\u003cp>\u003cstrong>50. Find points on the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> at which the tangents are\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i)\u003c/strong> Parallel to x-axis \u003cstrong>(ii)\u003c/strong> Parallel to y-axis.\u003c/p>","answerText":"\u003cp>Diffrentiating \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> wrt. X we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(i) \u003c/strong>When the tangent is || to x-axis, the slope of tangent is 0\u003c/p>\u003cp>ie, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> putting this in the eq\u003csup>n\u003c/sup> of curve. We get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The point at which the tangents are parallel to x-axis are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)and \u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> When the tangent is parallel to y-axis, the slope of the normal is 0.\u003c/p>\u003cp>ie, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> , putting this in the eq\u003csup>n\u003c/sup> of curve we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The point at which the tangents are parallel to y-axis are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)and \u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":61,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-50-find-points-on-the-curve-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-qna-11425402"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398466,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425418,"answerId":11425424,"questionText":"\u003cp>\u003cstrong>51. Find the equations of the tangent and normal to the given curves at the indicated points:\u003c/strong>\u003c/p>\n\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmi>v\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>v\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>t\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>we have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence eq\u003csup>n\u003c/sup> of tangent at (0, 5) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And eq\u003csup>n\u003c/sup> of normal at (0, 5) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 30 28\u003c/p>\u003cp>= 2\u003c/p>\u003cp>Slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence eq\u003csup>n\u003c/sup> of tangent at (1, 3) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And eq\u003csup>n\u003c/sup> of normal at (1,3) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iii)\u003c/strong> We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, eqn of tangent at (1, 1) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And eq\u003csup>n\u003c/sup> of normal at (1,1) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, eq\u003csup>n\u003c/sup> of the tangent at (0,0) is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>ie, x- axis\u003c/p>\u003cp>Hence, the eq\u003csup>n\u003c/sup> of normal is x = 0 ie, y-axis\u003c/p>\u003cp>\u003cstrong>(v) \u003c/strong>We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, slope of tangent \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>/\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1733994843phph85dog_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1733994843phph85dog.jpeg\" alt=\"\" width=\"533\" height=\"300\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1733994858phpDieBA4_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1733994858phpDieBA4.jpeg\" alt=\"\" width=\"288\" height=\"204\">\u003c/picture>\u003c/div>\u003c/div>","order":62,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-51-find-the-equations-of-the-tangent-and-normal-to-the-given-curves-at-the-indicated-points-qna-11425418"},"anaFaqSource":"manual","generationSource":"manual","templateId":143,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398468,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425438,"answerId":11425446,"questionText":"\u003cp>\u003cstrong>52. Find the equation of the tangent line to the curve y = x\u003csup>2\u003c/sup> - 2x +7 which is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(a)\u003c/strong> Parallel to the line 2x - y + 9 = 0 \u003cstrong>(b)\u003c/strong> Perpendicular to the line 5y - 15x = 13\u003c/p>","answerText":"\u003cp>The eq\u003csup>n\u003c/sup>of the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(a)\u003c/strong> The line \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> compared to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> gives,\u003c/p>\u003cp>Slope of line = 2.\u003c/p>\u003cp>If the tangent of the curve is parallel to the line\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=slope of line\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the point of contact of the tangent is (2, 7)\u003c/p>\u003cp>The eq\u003csup>n\u003c/sup> of tangent is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(b) \u003c/strong>The line \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>compared to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> gives\u003c/p>\u003cp>slope of line = 3\u003c/p>\u003cp>As the tangent to the curve is ⊥ to the line.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>= -1/slope opf line\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> we get \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the point of contact of the tangent is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And eq\u003csup>n\u003c/sup> of the tangent is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":63,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-52-find-the-equation-of-the-tangent-line-to-the-curve-y-amp-nbsp-x-lt-sup-gt-2-lt-sup-gt-qna-11425438"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398470,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425470,"answerId":11425472,"questionText":"\u003cp>\u003cstrong>53. Show that the tangents to the curve y = 7x\u003csup>3\u003c/sup> + 11 at the points where x = 2 and x = - 2 are parallel.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>Slope of tangent \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The tangent to the given curve at x = 2 and x = -2 are parallel.\u003c/p>","order":64,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-53-show-that-the-tangents-to-the-curve-y-7x-lt-sup-gt-3-lt-sup-gt-11-at-the-points-where-qna-11425470"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398472,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425482,"answerId":11425486,"questionText":"\u003cp>\u003cstrong>54. Find the points on the curve y = x\u003csup>3\u003c/sup> at which the slope of the tangent is equal to the y-coordinate of the point.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As, slope of tangent = y - coordinate of the point.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and when \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmo>, \u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The required points are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)and \u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":65,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-54-find-the-points-on-the-curve-y-x-lt-sup-gt-3-lt-sup-gt-at-which-the-slope-of-the-tangent-qna-11425482"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398474,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425490,"answerId":11425494,"questionText":"\u003cp>\u003cstrong>55. For the curve y = 4x\u003csup>3\u003c/sup> - 2x\u003csup>5\u003c/sup> , find all the points at which the tangent \u003c/strong>\u003cstrong>passes through the origin.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Slope of tangent, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> ________(1)\u003c/p>\u003cp>Let P(x, y) be the required point at the tangent passing through the origin (0,0)\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> _________(2)\u003c/p>\u003cp>So, from (1) and (2) we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Putting this value of y in the eq\u003csup>n\u003c/sup> of curve we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The required points are (0,0), (1,2) and (1,2)\u003c/p>","order":66,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-55-for-the-curve-y-4x-lt-sup-gt-3-lt-sup-gt-ndash-2x-lt-sup-gt-5-lt-sup-gt-find-all-the-points-qna-11425490"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398476,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425504,"answerId":11425506,"questionText":"\u003cp>\u003cstrong>56. Find the points on the curve x\u003csup>2\u003c/sup> + y\u003csup>2\u003c/sup> - 2x - 3 = 0 at which the tangents are parallel to the x-axis.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup>of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> ________ (1)\u003c/p>\u003cp>Differentiating the given curve wrt.x we get, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> slope of tangent\u003c/p>\u003cp>Given, tangent is | to x-axis\u003c/p>\u003cp>ie, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Putting x = 1 in eq\u003csup>n\u003c/sup> (1) we get, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>i\u003c/mi>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the required points are (1,2) and (1, 2).\u003c/p>","order":67,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-56-find-the-points-on-the-curve-x-lt-sup-gt-2-lt-sup-gt-y-lt-sup-gt-2-lt-sup-gt-ndash-2x-qna-11425504"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398478,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425510,"answerId":11425516,"questionText":"\u003cp>\u003cstrong>57. Find the equation of the normal at the point (am\u003csup>2\u003c/sup> ,am\u003csup>3\u003c/sup> ) for the curve ay\u003csup>2\u003c/sup> = x\u003csup>3\u003c/sup>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> _____(1)\u003c/p>\u003cp>Differentiating eq\u003csup>n\u003c/sup> (1) wrt.x we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> , slope of tangent\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>corresponding slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, eq\u003csup>n\u003c/sup> of normal at \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>m\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":68,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-57-find-the-equation-of-the-normal-at-the-point-am-lt-sup-gt-2-lt-sup-gt-am-lt-sup-gt-3-lt-qna-11425510"},"anaFaqSource":"manual","generationSource":"manual","templateId":167,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398480,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425518,"answerId":11425522,"questionText":"\u003cp>\u003cstrong>58. Find the equation of the normals to the curve y = x\u003csup>3\u003c/sup> + 2x + 6 which are \u003c/strong>\u003cstrong>parallel to the line x + 14y + 4 = 0.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> _____(1)\u003c/p>\u003cp>slope of tangent to the curve, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>so, slope of normal to the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, the line \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> compared to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> gives\u003c/p>\u003cp>slope of line = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As the normal is parallel to the line\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and when \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The point of contact of the normal are (2, 18) and (-2, -6)\u003c/p>\u003cp>Hence the eq\u003csup>n\u003c/sup> of normal are\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":69,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-58-find-the-equation-of-the-normals-to-the-curve-y-x-lt-sup-gt-3-lt-sup-gt-2x-6-which-are-qna-11425518"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398482,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425526,"answerId":11425530,"questionText":"\u003cp>\u003cstrong>59. Find the equations of the tangent and normal to the parabola y\u003csup>2\u003c/sup> = 4ax at the point (at\u003csup>2\u003c/sup> , 2at).\u003c/strong>\u003c/p>","answerText":"\u003cp>The given of the parabola is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>slope of tangent is given by\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>so, slope of normal\u003c/p>\u003cp>so, slope of normal \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence eq\u003csup>n\u003c/sup> of tangent at point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And eq\u003csup>n\u003c/sup> of normal at point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>t\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":70,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-59-find-the-equations-of-the-tangent-and-normal-to-the-parabola-y-lt-sup-gt-2-lt-sup-gt-4ax-qna-11425526"},"anaFaqSource":"manual","generationSource":"manual","templateId":143,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398484,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425534,"answerId":11425536,"questionText":"\u003cp>\u003cstrong>60. Prove that the curves x = y\u003csup>2 \u003c/sup>and xy = k cut at right angles* if 8k\u003csup>2\u003c/sup> = 1.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n\u003c/sup> of the curves are\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> ________(1)\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> ___________(2)\u003c/p>\u003cp>Differentiating eqn (1) and (2) wrt ‘x’ we get,\u003c/p>\u003cp>\u003cmath>\u003c/math> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> _________(3)\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> _________(4)\u003c/p>\u003cp>Since the two curves cut each other at right angles we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\times\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> ________(5)\u003c/p>\u003cp>The point of intersection can be solve from eqn (1) and (2), \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>k\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>k\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>/\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>/\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, using eq\u003csup>n\u003c/sup> (5) we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>/\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>/\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>8\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence proved\u003c/p>","order":71,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-60-prove-that-the-curves-x-y-lt-sup-gt-2-lt-sup-gt-and-xy-k-cut-at-right-angles-if-8k-lt-qna-11425534"},"anaFaqSource":"manual","generationSource":"manual","templateId":141,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398486,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425540,"answerId":11425542,"questionText":"\u003cp>\u003cstrong>61. Find the equations of the tangent and normal to the hyperbola\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> at the point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eq\u003csup>n \u003c/sup>of the hyperbola is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> ______(1)\u003c/p>\u003cp>Differentiating eqn (1) wrt ‘x’ we get,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is the reqd slope of tangent to the curve\u003c/p>\u003cp>So, eq\u003csup>n \u003c/sup>of tangent at point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> lies on the parabola given by eqn (1) we write,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":72,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-61-find-the-equations-of-the-tangent-and-normal-to-the-hyperbola-lt-strong-gt-lt-p-gt-lt-qna-11425540"},"anaFaqSource":"manual","generationSource":"manual","templateId":119,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398492,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425568,"answerId":11425574,"questionText":"\u003cp>\u003cstrong>64. The line y = x + 1 is a tangent to the curve y\u003csup>2\u003c/sup> = 4x at the point (A) (1, 2) (B) (2, 1) (C) (1, - 2) (D) (- 1, 2)\u003c/strong>\u003c/p>","answerText":"\u003cp>The given eqn of curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>Then, differentiating wrt x we get, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> which is the slope of the tangent to the curve.\u003c/p>\u003cp>The line \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> compared to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> gives slope of line = 1.\u003c/p>\u003cp>Since, tangent is the line we have, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Putting y = 2 in \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> we get, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the required point is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Option (A) is correct.\u003c/p>","order":73,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-64-the-line-y-x-1-is-a-tangent-to-the-curve-amp-nbsp-y-lt-sup-gt-2-lt-sup-gt-4x-at-the-point-qna-11425568"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398494,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425590,"answerId":11425602,"questionText":"\u003cp>\u003cstrong>65. Using differentials, find the approximate value of each of the following up to 3 places of decimal.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>Let y= ?x\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> : Let x = 25 and x = 0. 3.\u003c/p>\u003cp>Then, ?y = ?x+?x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??x\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?25.3\u003c/mo>\u003cmo>??25\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?25.3\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??25.3\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmtext>?\u003c/mtext>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>~\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 5 + dy\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?x\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?25\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 5 + 0.03\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??25.3\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> ?49.5\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>A.(ii)\u003c/p>\u003cp>Let y = ?x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 49 and x = 0.5.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=?x+?x\u003c/mo>\u003cmo>??x\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?49.5\u003c/mo>\u003cmo>??49\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??49.5\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsubsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msubsup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?x\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?49\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 7 + 0.0357.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??49.5\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iii) \u003c/strong>?0.6\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>A.(iii)\u003c/p>\u003cp>Let y = ?x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 0.64 and ?x = 0.04.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=?x+Ax\u003c/mo>\u003cmo>??x\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?0.64-0.04\u003c/mo>\u003cmo>??0.64\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?0.6\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??0.6=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?x\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?0.64\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmtext>?\u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 0.8 - 0.025.\u003c/p>\u003cp>= 0.775.\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(iv)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> Let x = 0.008 and ?x = 0.00 1.\u003c/p>\u003cp>Then, ?y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 0.2 + 0.0083.\u003c/p>\u003cp>= 0.208.\u003c/p>\u003cp>\u003cstrong>(v) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(v)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 1 and ?x = -0.001\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>A\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>.\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 0.999.\u003c/p>\u003cp>\u003cstrong>(vi)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(vi)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Then, x = 16 and ?x = 1.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>A\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(vii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(vii)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 27 and ?x = 1.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">F)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(viii) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(viii)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 256 and ?x = 1.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(ix) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(ix)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> let x = 81 and ?x = 1.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(x)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(x)\u003c/p>\u003cp>Let y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = ?x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 400 and ?x = 1.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmtext>?\u003c/mtext>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmtext>?\u003c/mtext>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmtext>?\u003c/mtext>\u003cmo>+\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?x\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?400\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(xi) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(xi)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> Let x = 0.0036 and ?x = 0.0001\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=?x+?x\u003c/mo>\u003cmo>??x\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=?0.0037\u003c/mo>\u003cmo>??0.0036\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>??0.0037\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>D\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2?x\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2?0.0036\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 0.06 + 0.0008.\u003c/p>\u003cp>= 0.0608.\u003c/p>\u003cp>\u003cstrong>(xii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(xii)\u003c/p>\u003cp>Let \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Let x = 27 and ?x = 0.43.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>?\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>A\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmi>?\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmo st","order":74,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-65-using-differentials-find-the-approximate-value-of-each-of-the-following-up-to-3-places-qna-11425590"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398498,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425626,"answerId":11425628,"questionText":"\u003cp>\u003cstrong>67. Find the approximate value of f (5.001), where f(x) = x\u003csup>3\u003c/sup> - 7x2 + 15.\u003c/strong>\u003c/p>","answerText":"\u003cp>Given, y = f (x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 7\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 15.\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> dy = (3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 14\u003cem>x\u003c/em>) dx.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Δy = (3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 14\u003cem>x\u003c/em>) Δx.\u003c/p>\u003cp>Let, x = 5 and Δx = 0.001. Then, \u003c/p>\u003cp>Δy = f (x + Δx) f (x).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x + Δx) = f (x) + Δy = f (x) + (3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 4\u003cem>x\u003c/em>) Δx.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (5 + 0.001) = 5\u003csup>3\u003c/sup>- 7 (5)\u003csup>2\u003c/sup> + 15 + [3 (5)\u003csup>2\u003c/sup> - 14 (5)]. (0.001).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (5.001) = 125 - 175 + 15 + (75 - 70) (0.001)\u003c/p>\u003cp>= -35 + 0.005 = - 34.995.\u003c/p>","order":75,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-67-find-the-approximate-value-of-f-5-001-where-f-x-x-lt-sup-gt-3-lt-sup-gt-ndash-7x2-15-lt-qna-11425626"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398502,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425648,"answerId":11425652,"questionText":"\u003cp>\u003cstrong>69. Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%\u003c/strong>\u003c/p>","answerText":"\u003cp>We know that, the surface area 5 of a ‘x’ when length cube, is S = 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Given decrease in side, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>%\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo> \u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":76,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-69-find-the-approximate-change-in-the-surface-area-of-a-cube-of-side-x-metres-caused-by-decreasing-qna-11425648"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398504,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425658,"answerId":11425660,"questionText":"\u003cp>\u003cstrong>70. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x be the radius of the sphere & Δπ be the error in measuring the radius.\u003c/p>\u003cp>Then, π = 7m and Δr = 0.02m.\u003c/p>\u003cp>Now, volume v of sphere is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>V\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dU\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\cdot\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>Δ\u003c/mi>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> dv = 4π (7)\u003csup>2\u003c/sup> (.0.02) = 3.92 πm3\u003c/p>\u003cp>∴The appropriate error is calculating the volume is 3.92π\u003cem>m\u003c/em>\u003csup>3\u003c/sup>.\u003c/p>","order":77,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-70-if-the-radius-of-a-sphere-is-measured-as-7-m-with-an-error-of-0-02-m-then-find-the-approximate-qna-11425658"},"anaFaqSource":"manual","generationSource":"manual","templateId":167,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398506,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425664,"answerId":11425666,"questionText":"\u003cp>\u003cstrong>71. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let be the radius of the sphere &r be the error in measuring the radius.\u003c/p>\u003cp>Then, π = 9m and Δr = 0.03m.\u003c/p>\u003cp>Now, surface area S of the sphere is\u003c/p>\u003cp>S = 4π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴e, this = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> Δr = 8πr.Δr = 8π \times 9 \times 0.03\u003c/p>\u003cp>= 2.16π\u003cem>m\u003c/em>\u003csup>3\u003c/sup>.\u003c/p>\u003cp>Appropriate error in calculating the surface area is 2.16π\u003cem>m\u003c/em>\u003csup>3\u003c/sup>.\u003c/p>","order":78,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-71-if-the-radius-of-a-sphere-is-measured-as-9-m-with-an-error-of-0-03-m-then-find-the-approximate-qna-11425664"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398508,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425674,"answerId":11425678,"questionText":"\u003cp>\u003cstrong>72. If f(x) = 3x\u003csup>2\u003c/sup> + 15x + 5, then the approximate value of f (3.02) is (A) 47.66 (B) 57.66 (C) 67.66 (D) 77.66\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, y = f (x) = 3x\u003csup>2\u003c/sup> + 15x + 5.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> dy = (6x + 15) dx\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Δy = (6x + 15) Δx.\u003c/p>\u003cp>Let, x = 3 and Δx = 0.02 then, \u003c/p>\u003cp>Δy = f (x + Δx) - f (x)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x + Δx) = f (x) + Δy = f (x) + (6x + 5) Δx.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (3 + 0.02) = 3 (3)\u003csup>2\u003c/sup> + 15 (3) + 5 + (6 \times 3 + 15) (0.02).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (3.02) = 27 + 45 + 5 + (18 + 15) (0.02).\u003c/p>\u003cp>= 77 + 0.66\u003c/p>\u003cp>= 77.66\u003c/p>\u003cp>∴ Option (D) is correct.\u003c/p>","order":79,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-72-if-f-x-3x-lt-sup-gt-2-lt-sup-gt-15x-5-then-the-approximate-value-of-f-3-02-is-a-47-66-qna-11425674"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398510,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425682,"answerId":11425688,"questionText":"\u003cp>\u003cstrong>73. The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is (A) 0.06 x\u003csup>3\u003c/sup> m\u003csup>3\u003c/sup> (B) 0.6 x\u003csup>3 \u003c/sup>m\u003csup>3 \u003c/sup> (C) 0.09 x\u003csup>3\u003c/sup> m\u003csup>3\u003c/sup> (D) 0.9 x\u003csup>3\u003c/sup> m\u003csup>3\u003c/sup>\u003c/strong>\u003c/p>","answerText":"\u003cp>The volume v of a cube with side ‘x’ metre is v = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>Δ\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴increase in side, Δx = 3% of = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴dv = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>π \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo> m\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, option (C) is correct.\u003c/p>","order":80,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-73-the-approximate-change-in-the-volume-of-a-cube-of-side-x-metres-caused-by-increasing-the-qna-11425682"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398514,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426836,"answerId":11426848,"questionText":"\u003cp>\u003cstrong>75. Find the maximum and minimum values, if any, of the following functions \u003c/strong>\u003cstrong>given by\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003c/strong>f(x) = |x + 2| - 1\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>we have, f(x) = |x + 2| - 1\u003c/p>\u003cp>We know that, for all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>\geq\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x)\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\geq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>- 1.\u003c/p>\u003cp>∴ Minimum value of f(x) = -1 when x + 2 = 0 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = - 2.\u003c/p>\u003cp>And maximum value of f(x) does not exist.\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(ii)\u003c/p>\u003cp>We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>For all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>\leq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> g(x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003cmn>\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>3.\u003c/p>\u003cp>∴ Maximum value of g(x) = 3 when \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And minimum value does not exist.\u003c/p>\u003cp>\u003cstrong>(iii)\u003c/strong> h(x) = sin (2x) + 5.\u003c/p>\u003cp>A(iii)\u003c/p>\u003cp>we have, h(x) = sin (2x) + 5.\u003c/p>\u003cp>For all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> {range of sine function is [-1, 1]}\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -1 + 5 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>sin 2x + 5 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>1 + 5.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 4 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>h(x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>6.\u003c/p>\u003cp>∴ Maximum value of h(x) = 6.\u003c/p>\u003cp>Minimum value of h(x) = 4.\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(iv)\u003c/p>\u003cp>we have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As for all \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>?\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> -1 + 3 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>sin 4x + 3\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 1 + 3\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>|\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>\leq\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>\leq\u003c/mo>\u003cmo>|\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>|\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 2 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>f(x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\leq\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>4.\u003c/p>\u003cp>∴ Maximum value of f(x) = 4.\u003c/p>\u003cp>Minimum value of f(x) = 2.\u003c/p>\u003cp>\u003cstrong>(v)\u003c/strong> h(x) = x + 1.,x\in ( -1, 1).\u003c/p>\u003cp>A(v)\u003c/p>\u003cp>we have, h(x) = x + 1.,x\in ( -1, 1).\u003c/p>\u003cp>Given, -1<x<1< strong=\"\"></x<1<>\u003c/p>\u003cp>- 1 + 1 <\u003cem>x\u003c/em> + 1 < 1 + 1\u003c/p>\u003cp>0 <\u003cem>h\u003c/em> (\u003cem>x\u003c/em>) < 2.\u003c/p>\u003cp>∴ Maximum value and minimum value of h(x) does not exist.\u003c/p>","order":81,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-75-find-the-maximum-and-minimum-values-if-any-of-the-following-functions-lt-strong-gt-lt-qna-11426836"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398516,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426854,"answerId":11426872,"questionText":"\u003cp>\u003cstrong>76. Find the local maxima and local minima, if any, of the following functions.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong> Find also the local maximum and the local minimum values, as the case may be:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) f(x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>we have, f(x) = \u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>f(x) = 2x.\u003c/p>\u003cp>andf(x) = 2.\u003c/p>\u003cp>AR extreme point, f(x) = 0\u003c/p>\u003cp>2x = 0\u003c/p>\u003cp>x = 0.\u003c/p>\u003cp>When x = 0, f(0) = 2 > 0.\u003c/p>\u003cp>∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0\u003csup>2\u003c/sup> = 0.\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> 3x\u003c/p>\u003cp>A(ii)\u003c/p>\u003cp>we have, g(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 3\u003cem>x\u003c/em>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> g'(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> g''(x) = 6x.\u003c/p>\u003cp>At extreme point,\u003c/p>\u003cp>g'(x) = 0\u003c/p>\u003cp>3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3 = 0.\u003c/p>\u003cp>3(\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 1) = 0 \Rightarrow 3(x - 1)(x + 1) = 0.\u003c/p>\u003cp>x = 1 or x = -1.\u003c/p>\u003cp>At x = 1, g\"(1) = 6.1 = 6 > 0.\u003c/p>\u003cp>So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1\u003csup>3\u003c/sup>- 3.1 = 1 - 3 = - 2.\u003c/p>\u003cp>And at x = -1, g\"( -1) = 6 ( -1) = 6 < 0.\u003c/p>\u003cp>So, x = -1 is a point of local minima and value of local minimum is given by\u003c/p>\u003cp>g(- 1) = (- 1)\u003csup>3\u003c/sup>- 3(- 1) = 1 + 3 = 2.\u003c/p>\u003cp>\u003cstrong>(iii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(iii)\u003c/p>\u003cp>we have, h(x) = sin x + cos x.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> h'(x) = cos x - sin x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> h'(x) = -sin x - cos x.\u003c/p>\u003cp>At extreme points,\u003c/p>\u003cp>h'(x) = 0\u003c/p>\u003cp>cosx - sin x = 0\u003c/p>\u003cp>cosx = sin x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 1 = tan x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W.jpeg\" alt=\"\" width=\"502\" height=\"132\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cstrong>(iv) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(iv)\u003c/p>\u003cp>we have, f(x) = sin x cos x.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) = cos x + sin x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) = sin x + cos x.\u003c/p>\u003cp>At extreme points,\u003c/p>\u003cp>f(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> cosx + sin x = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> IInd quadrate or \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> IIIth quadrate\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>A\u003c/mi>\u003cmi>t\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003cspan title=\"Click to copy mathml\">\u003cbr>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn.jpeg\" alt=\"\" width=\"426\" height=\"371\">\u003c/picture>\u003c/div>\u003cdiv>\u003cstrong>(v)\u003c/strong> f(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 9\u003cem>x\u003c/em> + 15.\u003c/div>\u003c/div>\u003cp>A(v)\u003c/p>\u003cp>we have, f(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 9\u003cem>x\u003c/em> + 15.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 9.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f'(x) = 6x - 12.\u003c/p>\u003cp>At extreme point, f'(x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 9 = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 4\u003cem>x\u003c/em> + 3 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>-\u003cem>x \u003c/em>- 3\u003cem>x\u003c/em> + 3 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x(x - 1) -3(x - 1) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (x - 1)(x - 3) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 1 or x = 3.\u003c/p>\u003cp>At, x = 1,f\"(1) = 6 \times 1 - 12 = 6 - 12 = - 6 < 0\u003c/p>\u003cp>∴x = 1 is a point of local maxima and the value of local\u003c/p>\u003cp>Maximum is given by f(1) = 1\u003csup>3\u003c/sup>- 6(1)\u003csup>2\u003c/sup> + 9(1) + 15\u003c/p>\u003cp>= 1 - 6 + 9 + 15\u003c/p>\u003cp>= 19.\u003c/p>\u003cp>And at x = 3,f\"(3) = 6 \times 3 - 12 = 18 - 12 = 6 > 0\u003c/p>\u003cp>∴x = 3 is a point of local minima and the value of\u003c/p>\u003cp>local minimum is given by f (3) = 3\u003csup>3\u003c/sup>- 6(3)\u003csup>2\u003c/sup> + 9(3) + 15.\u003c/p>\u003cp>= 27 - 54 + 27 + 15.\u003c/p>\u003cp>= 15.\u003c/p>\u003cp>\u003cstrong>(vi)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(vi)\u003c/p>\u003cp>We have, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>g\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003cmi>z\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, extreme point, g'(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = -2 or x = 2\u003c/p>\u003cp>Given, that x > 0, hence we have x = 2.\u003c/p>\u003cp>At, x = 2, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmover accent=\"true\">\u003cmi>g\u003c/mi>\u003cmo>?\u003c/mo>\u003c/mover>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴x = 2 is a point of local minima and value of local\u003c/p>\u003cp>Minimum is given by g(2) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(vii)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A(vii)\u003c/p>\u003cp>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At extreme points, g'(x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 0.\u003c/p>\u003cp>At, x = 0, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>g\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>'\u003c/mo>\u003cmo>'\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴x = 0 is a point of local maxima and value of local maximum is given by \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0.jpeg\" alt=\"\" width=\"319\" height=\"295\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734069223phptw07x9.jpeg\" alt=\"\" width=\"389\" height=\"462\">\u003c/picture>\u003c/div>\u003c/div>","order":82,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-76-find-the-local-maxima-and-local-minima-if-any-of-the-following-functions-lt-strong-gt-qna-11426854"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398520,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426896,"answerId":11426902,"questionText":"\u003cp>\u003cstrong>78. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003c/strong>f(x) =\u003cem>x\u003c/em>\u003csup>3\u003c/sup> , x \in [- 2, 2]\u003c/p>","answerText":"\u003cp>\u003cstrong>(i) \u003c/strong>We have,\u003c/p>\u003cp>f(x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> , x \in [- 2, 2].\u003c/p>\u003cp>f(x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>At, f(x) = 0\u003c/p>\u003cp>3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 0\u003c/p>\u003cp>x = 0 <--[-2, 2].\u003c/p>\u003cp>We shall absolute the value of f at x = 0 and points of interval [ -2, 2]. So,\u003c/p>\u003cp>f(0) = 0\u003c/p>\u003cp>\u003cem>f\u003c/em>(- 2) = (- 2)\u003csup>3\u003c/sup> = 8\u003c/p>\u003cp>\u003cem>f\u003c/em>(2) = 2\u003csup>3\u003c/sup> = 8.\u003c/p>\u003cp>∴ Absolute maximum value of f(x) = 8 at x = 2\u003c/p>\u003cp>and absolute minimum value of f(x) = -8 at x = -2.\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> f (x) = sin x + cos x , x \in [0, π]\u003c/p>\u003cp>A.(ii)\u003c/p>\u003cp>We have, f(x) = sin x + cos x , x \in [0, π]\u003c/p>\u003cp>f(x) = cos x - sin x.\u003c/p>\u003cp>atf(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> cosx - sin x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> sinx = cos x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>°\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734072978phpTrh5em_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734072978phpTrh5em.jpeg\" alt=\"\" width=\"238\" height=\"109\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cstrong>(iii)\u003c/strong> f(x) = 4x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>A.(iii)\u003c/p>\u003cp>We have, f(x) = 4x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = 4 - x\u003c/p>\u003cp>atf(x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 4- x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 4 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>-2,\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 7.87.5\u003c/p>\u003cp>Hence, absolute maximum value of f(x) = 8 at x = 4\u003c/p>\u003cp>and absolute minimum value of f(x) = -10 at x = -2.\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3, \u003cem>x\in\u003c/em>[ -3, 1]\u003c/p>\u003cp>A.(iv)\u003c/p>\u003cp>We have, f(x) = (\u003cem>x \u003c/em>- 1)\u003csup>2\u003c/sup> + 3, \u003cem>x\in\u003c/em>[ -3, 1]\u003c/p>\u003cp>f(x) = 2(x -1)\u003c/p>\u003cp>atf(x) = 0\u003c/p>\u003cp>2(x - 1) = 0\u003c/p>\u003cp>x = 1\u003cem>\in\u003c/em> [ -3, 1]\u003c/p>\u003cp>∴f(1) = (1 - 1)\u003csup>2\u003c/sup> + 3 = 3\u003c/p>\u003cp>\u003cem>f\u003c/em>(- 3) = (- 3 - 1)\u003csup>2\u003c/sup> + 3 = 16 + 3 = 1\u003c/p>\u003cp>∴ Absolute maximum value of f(x) = 19 at x = -3.\u003c/p>\u003cp>And absolute minimum value of f(x) = 3 at x = 1.\u003c/p>","order":83,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-78-find-the-absolute-maximum-value-and-the-absolute-minimum-value-of-the-following-functions-qna-11426896"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398522,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426912,"answerId":11426918,"questionText":"\u003cp>\u003cstrong>79. Find the ma pro\u003c/strong>ximum\u003cstrong>fit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x\u003csup>2\u003c/sup>\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, p (x) = 41 -\u003cem>f\u003c/em>2\u003cem>x \u003c/em>- 18\u003cem>x\u003c/em>\u003csup>2\u003c/sup>.\u003c/p>\u003cp>P (x) = - 72 - 36\u003cem>x\u003c/em>\u003c/p>\u003cp>P (x) = -36\u003c/p>\u003cp>At extreme point, \u003c/p>\u003cp>- 72 - 36\u003cem>x\u003c/em> = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>At \u003cem>x\u003c/em> = - 2, \u003cem>p\"\u003c/em> (\u003cem>x\u003c/em>) = - 36 < 0.\u003c/p>\u003cp>∴x = -2 is a point of local maximum and the value of local\u003c/p>\u003cp>Maximum is given by P (2) = 41 - 72 (- 2) - 18 (- 2)\u003csup>2\u003c/sup>\u003c/p>\u003cp>41 + 144 - 72 = 113 units.\u003c/p>","order":84,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-79-find-the-ma-pro-lt-strong-gt-ximum-lt-strong-gt-fit-that-a-company-can-make-if-the-profit-qna-11426912"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398524,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426926,"answerId":11426934,"questionText":"\u003cp>\u003cstrong>80. Find both the maximum value and the minimum value of 3x\u003csup>4\u003c/sup> - 8x\u003csup>3\u003c/sup> + 12x\u003csup>2\u003c/sup> - 48x + 25 on the interval [0, 3].\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = 3\u003cem>x\u003c/em>\u003csup>4\u003c/sup>- 8\u003cem>x\u003c/em>\u003csup>3\u003c/sup> + 12\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 48\u003cem>x\u003c/em> + 25, \u003cem>x \in\u003c/em> [0, 3].\u003c/p>\u003cp>f (x) = 12\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 24\u003cem>x \u003c/em>- 48.\u003c/p>\u003cp>At f (x) = 0.\u003c/p>\u003cp>12\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 24\u003cem>x \u003c/em>- 48 = 0.\u003c/p>\u003cp>\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 2\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2\u003cem>x \u003c/em>- 4 = 0\u003c/p>\u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup> (\u003cem>x \u003c/em>- 2) + 2 (\u003cem>x \u003c/em>- 2) = 0\u003c/p>\u003cp> (\u003cem>x \u003c/em>- 2) + (\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2) = 0\u003c/p>\u003cp>x = 2 \in [0, 3] or x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\pm\sqrt-2\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> which is not possible as\u003c/p>\u003cp>∴f (x) = 3 (2)\u003csup>4\u003c/sup>- 8 (2)\u003csup>3\u003c/sup> + 12 (2)\u003csup>2\u003c/sup>- 4 (2) + 25.\u003c/p>\u003cp>=48 - 64 + 48 - 96 + 25.\u003c/p>\u003cp>= -39.\u003c/p>\u003cp>f (0) =3 (0)\u003csup>4\u003c/sup>- 8 (0)\u003csup>3\u003c/sup> + 12 (0)\u003csup>2\u003c/sup>- 48 (0) + 25.\u003c/p>\u003cp>= 25.\u003c/p>\u003cp>f (3) = 3 (3)\u003csup>4\u003c/sup>- 8 (3)\u003csup>3\u003c/sup> + 12 (3)\u003csup>2\u003c/sup>- 48 (3) + 25.\u003c/p>\u003cp>= 243 - 216 + 108 - 144 + 25\u003c/p>\u003cp>= 16.\u003c/p>\u003cp>Maximum value of f (x) = 25 at x = 0.\u003c/p>\u003cp>and minimum value of f (x) = -39 at x = 2.\u003c/p>","order":85,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-80-find-both-the-maximum-value-and-the-minimum-value-of-3x-lt-sup-gt-4-lt-sup-gt-ndash-8x-qna-11426926"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398526,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426940,"answerId":11426944,"questionText":"\u003cp>\u003cstrong>81. At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f(x) = sin 2x, x \in [0, 2π],\u003c/p>\u003cp>f(x) = 2cos 2x.\u003c/p>\u003cp>At f(x) = 0.\u003c/p>\u003cp>2 cos 2x = 0\u003c/p>\u003cp>cos 2x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmo>\in\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 1. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> = 1.\u003c/p>\u003cp>f(0) = sin 2(0) = sin 0 = 0\u003c/p>\u003cp>f(2π) = sin 2(2π) = sin 4π = 0\u003c/p>\u003cp>Hence, the points of maximum \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmi>f\u003c/mi>\u003cmi>x are.\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)and \u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":86,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-81-at-what-points-in-the-interval-does-the-function-sin-2x-attain-its-maximum-value-lt-strong-qna-11426940"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398530,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426984,"answerId":11426986,"questionText":"\u003cdiv>\n\u003cdiv>\u003cstrong>83. Find the maximum value of 2x\u003csup>3\u003c/sup> - 24x + 107 in the interval [1, 3].\u003c/strong>\u003c/div>\n\u003c/div>\n\u003cp>\u003cstrong>Find the maximum value of the same function in [-3, -1]\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) =2\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 24\u003cem>x\u003c/em> + 107, \u003cem>x\u003c/em> [1,3]\u003c/p>\u003cp>f (x) = 6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 24.\u003c/p>\u003cp>At f (x) = 0\u003c/p>\u003cp>6\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 24= 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = \pm2. ->x = 2 \in [1, 3].\u003c/p>\u003cp>So, f (2) = 2 (2)\u003csup>3\u003c/sup>- 24 (2) + 107 = 16 - 48 + 107 = 75.\u003c/p>\u003cp>f (1) = 2 (1)\u003csup>3\u003c/sup>- 24 (1) + 107 = 2 - 24 + 107 = 85.\u003c/p>\u003cp>f (3) = 2 (3)\u003csup>3\u003c/sup>- 24 (3) + 107 = 54 - 72 + 107 = 89\u003c/p>\u003cp>∴ Maximum value of f (x) in interval [1, 3] is 89 at x = 3.\u003c/p>\u003cp>When x \in [ -3, -1]\u003c/p>\u003cp>From f (x) = 0\u003c/p>\u003cp>x = -2 \in [ -3, -1]\u003c/p>\u003cp>So, \u003cem>f\u003c/em> (- 2) = 2 (- 2)\u003csup>3\u003c/sup>- 24 (- 2) + 107 = - 16 + 48 + 107 = 139.\u003c/p>\u003cp>\u003cem>f\u003c/em> (- 3) = 2 (- 3)\u003csup>3\u003c/sup>- 24 (- 3) + 107 = - 54 + 72 + 107 = 125.\u003c/p>\u003cp>\u003cem>f\u003c/em> (- 1) = 2 (- 1)\u003csup>3\u003c/sup>- 24 (- 1) + 107 = - 2 + 24 + 107 = 129.\u003c/p>\u003cp>∴ Maximum value of f (x) in interval [ -3, -1] is 139 at x = -2.\u003c/p>","order":87,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-div-gt-lt-div-gt-lt-strong-gt-83-find-the-maximum-value-of-2x-lt-sup-gt-3-lt-sup-gt-ndash-24x-107-in-the-interval-qna-11426984"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398532,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426994,"answerId":11427000,"questionText":"\u003cp>\u003cstrong>84. It is given that at x = 1, the function x\u003csup>4\u003c/sup> - 62x\u003csup>2\u003c/sup> + ax + 9 attains \u003c/strong>\u003cstrong>its maximum value, on the interval [0, 2]. Find the value of a.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = \u003cem>x\u003c/em>\u003csup>4\u003c/sup>- 62\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>ax\u003c/em> + 9, \u003cem>x\in\u003c/em> [0, 2].\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) = 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 124\u003cem>x\u003c/em> + \u003cem>a\u003c/em>\u003c/p>\u003cp>∴f (x) active its max\u003csup>n\u003c/sup> value at x = 1\u003cem>\in\u003c/em> [0, 2]\u003c/p>\u003cp>∴f (1) = 0.\u003c/p>\u003cp>4 (1)\u003csup>3\u003c/sup>- 124 (1) + a = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> a = 124 - 4 = 120.\u003c/p>\u003cp>∴a = 120\u003c/p>","order":88,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-84-it-is-given-that-at-x-1-the-function-x-lt-sup-gt-4-lt-sup-gt-ndash-62x-lt-sup-gt-2-lt-qna-11426994"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398536,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427038,"answerId":11427044,"questionText":"\u003cp>\u003cstrong>86. Find two numbers whose sum is 24 and whose product is as large as possible.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘x’ and ‘y’ be the two number\u003c/p>\u003cp>Then, x + y = 24 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> y = 24 - x\u003c/p>\u003cp>Let ‘P’ be their product then, \u003c/p>\u003cp>P = xy = x (24 - x) = 24\u003cem>x \u003c/em>-\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>P (x) = 24\u003cem>x \u003c/em>-\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>24 - 2x = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, P (12) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = 12 is a point of local maxima\u003c/p>\u003cp>Hence, y = 24 - 12 = 12.\u003c/p>\u003cp>The uqdtwno (x, y) is (12, 12).\u003c/p>","order":89,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-86-find-two-numbers-whose-sum-is-24-and-whose-product-is-as-large-as-possible-lt-strong-gt-qna-11427038"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398538,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427052,"answerId":11427056,"questionText":"\u003cp>\u003cstrong>87. Find two positive numbers x and y such that x + y = 60 and xy\u003csup>3\u003c/sup> is maximum.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, x + y = 60.where x, y > 0\u003c/p>\u003cp>x = 60 - y.\u003c/p>\u003cp>Let the product P = \u003cem>xy\u003c/em>\u003csup>3\u003c/sup> = (60 -\u003cem>y\u003c/em>) \u003cem>y\u003c/em>\u003csup>3\u003c/sup> = 60\u003cem>y\u003c/em>\u003csup>3\u003c/sup>-\u003cem>y\u003c/em>\u003csup>4\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 4\u003cem>y\u003c/em>\u003csup>2\u003c/sup> (45 -\u003cem>y\u003c/em>)\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 4\u003cem>y\u003c/em>\u003csup>2\u003c/sup> (45 -\u003cem>y\u003c/em>) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> y = 0 and y = 45\u003c/p>\u003cp>As y > 0, y = 45\u003c/p>\u003cp>When, y > 45, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= ve < 0\u003c/p>\u003cp>Ad y < 45, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= (+ ve) > 0\u003c/p>\u003cp>∴p is maximum when y = 45 from + ve to- ve or y increases through 45.\u003c/p>\u003cp>So, x = 60 - y = 6Ø - 45 = 15.\u003c/p>\u003cp>Øx = 15 and y = 45.\u003c/p>","order":90,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-87-find-two-positive-numbers-x-and-y-such-that-x-y-60-and-xy-lt-sup-gt-3-lt-sup-gt-is-maximum-qna-11427052"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398540,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427062,"answerId":11427076,"questionText":"\u003cp>\u003cstrong>88. Find two positive numbers x and y such that their sum is 35 and the product \u003c/strong>\u003cstrong>x\u003csup>2 \u003c/sup>y\u003csup>5 \u003c/sup> is a maximum.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, x + y = 35.\u003c/p>\u003cp>y = 35 - x\u003c/p>\u003cp>Let the product, P =x\u003csup>2 \u003c/sup>y\u003csup>5 \u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> P = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> (35 -\u003cem>x\u003c/em>)\u003csup>5\u003c/sup>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> 5 (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (1) + (35 -\u003cem>x\u003c/em>)\u003csup>5\u003c/sup> 2\u003cem>x\u003c/em>\u003c/p>\u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> [ - 5\u003cem>x\u003c/em> + (35 -\u003cem>x\u003c/em>) 2]\u003c/p>\u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> [ - 5\u003cem>x\u003c/em> + 70 - 2\u003cem>x\u003c/em>]\u003c/p>\u003cp>= \u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (70 - 7\u003cem>x\u003c/em>)\u003c/p>\u003cp>= 7\u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (10 -\u003cem>x\u003c/em>)\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>7\u003cem>x\u003c/em> (35 -\u003cem>x\u003c/em>)\u003csup>4\u003c/sup> (10 -\u003cem>x\u003c/em>) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 0, 35, 10\u003c/p>\u003cp>As x is a (+) ve number we have only\u003c/p>\u003cp>x = 10, 35\u003c/p>\u003cp>And again (at x = 35) y = 35 = 0 but yis also a (+) ve number\u003c/p>\u003cp>we get, x = 10 (only)\u003c/p>\u003cp>whenx < 10, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>? \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and when x > 10, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> changes from (+ ve) to ( -ve) as x increases while passing through 10\u003c/p>\u003cp>Hence, x = 10 is a point of local maxima\u003c/p>\u003cp>So, y = 35 - 10 = 25\u003c/p>\u003cp>∴x = 10 and y = 25\u003c/p>","order":91,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-88-find-two-positive-numbers-x-and-y-such-that-their-sum-is-35-and-the-product-lt-strong-qna-11427062"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398542,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427096,"answerId":11427102,"questionText":"\u003cp>\u003cstrong>89. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum\u003c/strong>\u003c/p>","answerText":"\u003cp>Let, x and y be the two positive number\u003c/p>\u003cp>Then, x + y = 16 y = 16 - x\u003c/p>\u003cp>Let p be the sun of the cubes then\u003c/p>\u003cp>p = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>y\u003c/em>\u003csup>3\u003c/sup> = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + (16 -\u003cem>x\u003c/em>)\u003csup>3\u003c/sup> = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + (16)\u003csup>3\u003c/sup>-\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 48\u003cem>x\u003c/em> (16 -\u003cem>x\u003c/em>)\u003c/p>\u003cp>p = 16\u003csup>3\u003c/sup> + 48\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 76 8\u003cem>x\u003c/em>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>96x - 768 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴at x = 8, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>p\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, x = 8 is a point of local minima\u003c/p>\u003cp>So, y = 16 - 8 = 8\u003c/p>\u003cp>Hence, x = 8, y = 8\u003c/p>","order":92,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-89-find-two-positive-numbers-whose-sum-is-16-and-the-sum-of-whose-cubes-is-minimum-lt-strong-qna-11427096"},"anaFaqSource":"manual","generationSource":"manual","templateId":91,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398544,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427120,"answerId":11427128,"questionText":"\u003cp>\u003cstrong>90. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.\u003c/strong>\u003c/p>","answerText":"\u003cp>Side of the tin square piece = 18 cm\u003c/p>\u003cp>Let x cm be the thought of the square to be cut from each corner.\u003c/p>\u003cp>The volume v of the box after cutting\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> v = length \times breadth \times height\u003c/p>\u003cp>= (18 - 2\u003cem>x\u003c/em>) (18 - 2\u003cem>x\u003c/em>) \u003cem>x \u003c/em>\u003cem>x\u003c/em>\u003c/p>\u003cp>= (18 - 2\u003cem>x\u003c/em>)\u003csup>2\u003c/sup>\u003cem>x\u003c/em>\u003c/p>\u003cp>= (324 + 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 72\u003cem>x\u003c/em>) \u003cem>x\u003c/em>\u003c/p>\u003cp>= 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 72\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 324\u003cem>x\u003c/em>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 12\u003cem>x\u003c/em> + 27 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 9\u003cem>x\u003c/em>- 3\u003cem>x\u003c/em> + 27 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em> (\u003cem>x \u003c/em>- 9) - 3 (\u003cem>x \u003c/em>- 9) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (\u003cem>x \u003c/em>- 9) (\u003cem>x \u003c/em>- 3) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 9 and x = 3\u003c/p>\u003cp>At x = 0, length of box = 18 - 2π9 = 18 - 18 = 0\u003c/p>\u003cp>Which is not possible\u003c/p>\u003cp>And at x = 3, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 24 (3) - 144 = -72 < 0\u003c/p>\u003cp>∴x = 3 is a point of maximum\u003c/p>\u003cp>Hence, ‘3’ cm (square) is to be cut from each side of the square\u003c/p>\u003cp>So that volume of box is maximum\u003c/p>","order":93,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-90-a-square-piece-of-tin-of-side-18-cm-is-to-be-made-into-a-box-without-top-by-cutting-a-qna-11427120"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398546,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427142,"answerId":11427148,"questionText":"\u003cp>\u003cstrong>91. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘x’ cm be the length of side of the square to be cut off from the rectangular surface\u003c/p>\u003cp>Then, the volume v of the box is v = (45 - 2\u003cem>x\u003c/em>) (24 - 2\u003cem>x\u003c/em>) \u003cem>x\u003c/em>\u003c/p>\u003cp>= 1080\u003cem>x \u003c/em>- 138\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 4\u003cem>x\u003c/em>\u003csup>3\u003c/sup>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 23\u003cem>x\u003c/em> + 90 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 5\u003cem>x \u003c/em>- 18\u003cem>x\u003c/em> + 90 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>x\u003c/em> (\u003cem>x \u003c/em>- 5) - 18 (\u003cem>x \u003c/em>- 5) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (\u003cem>x\u003c/em>- 5) (\u003cem>x\u003c/em>- 18) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> x = 5 and x = 18\u003c/p>\u003cp>At x = 18, breadth = 24 - 2 (18) = 24 - 36 = -12 which is not possible\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>, \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, x = 5 is the point of maximum\u003c/p>\u003cp>So, ‘5’ cm length of square seeds to be cut from each corner of the secthgle\u003c/p>","order":94,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-91-a-rectangular-sheet-of-tin-45-cm-by-24-cm-is-to-be-made-into-a-box-without-top-by-cutting-qna-11427142"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398550,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427178,"answerId":11427186,"questionText":"\u003cp>\u003cstrong>93. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r and h be the radius and height of the cylinder\u003c/p>\u003cp>So, r, h > 0\u003c/p>\u003cp>The total surface area s is given by\u003c/p>\u003cp>S = 2πr(h + r) = content .\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>r\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = content = x (say)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>u\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Then, the volume v of the cylinder\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>V\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>k\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>For maximum, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734082150php0lrnv8.jpeg\" alt=\"\" width=\"246\" height=\"266\">\u003c/picture>\u003c/div>\u003c/div>","order":95,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-93-show-that-the-right-circular-cylinder-of-given-surface-and-maximum-volume-is-such-that-qna-11427178"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398552,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427198,"answerId":11427202,"questionText":"\u003cp>\u003cstrong>94. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?\u003c/strong>\u003c/p>","answerText":"\u003cp>The volume v of a cylinder of height h and radius r is\u003c/p>\u003cp>V = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003cem>h\u003c/em> = 100\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Let, s be the surface area then\u003c/p>\u003cp>S = 2\u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003cem>h\u003c/em>r(r + h) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>S\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>r\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\RightarrowM\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>M\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2S\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> isa point of minimum\u003c/p>\u003cp>And \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":96,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-94-of-all-the-closed-cylindrical-cans-right-circular-of-a-given-volume-of-100-cubic-centimetres-qna-11427198"},"anaFaqSource":"manual","generationSource":"manual","templateId":91,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398554,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427216,"answerId":11427222,"questionText":"\u003cp>\u003cstrong>95. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x and y in ‘m’ be the length of side of the square the radius of the circle respectily\u003c/p>\u003cp>Then, length of wire = perimeter of square + circumference of circle\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 28 = 4x + 2πy\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 2x + πy = 14\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The combine area A of the square and the circle is\u003c/p>\u003cp>A = \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + π\u003cem>y\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2A\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dA\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>7\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2A\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>γ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> isa point of minima\u003c/p>\u003cp>Hence, length of square = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and length of circle = 2πy\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":97,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-95-a-wire-of-length-28-m-is-to-be-cut-into-two-pieces-one-of-the-pieces-is-to-be-made-into-qna-11427216"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398556,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427238,"answerId":11427244,"questionText":"\u003cp>\u003cstrong>96. Prove that the volume of the largest cone that can be inscribed in a sphere \u003c/strong>\u003cstrong>of radius R is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> of the volume of the sphere.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r and h be the radius and height of the one in scribed in the sphere of radius R.\u003c/p>\u003cp>Then, is ΔOBC, rt angle at B (\u003cem>h\u003c/em>-\u003cem>r\u003c/em>)\u003csup>2\u003c/sup> + \u003cem>r\u003c/em>\u003csup>2\u003c/sup> = R\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>h\u003c/em>\u003csup>2\u003c/sup> + R\u003csup>2\u003c/sup>- 2\u003cem>h\u003c/em>R + \u003cem>h\u003c/em>\u003csup>2\u003c/sup> = R\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>r\u003c/em>\u003csup>2\u003c/sup> = 2\u003cem>h\u003c/em>R -\u003cem>h\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>Then the volume v of the cone is, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>V\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003cmi>R\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>Rh\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>Rh\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2V\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4R\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>4R\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>4Rh - 3\u003cem>h\u003c/em>\u003csup>2\u003c/sup> = 0.\u003c/p>\u003cp>h(4R - 3h) = 0.\u003c/p>\u003cp>h = 0 and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>R\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As h> 0, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4R\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2v\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>R\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4rR\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>4R\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo stretchy=\"false\">R]\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>Rπ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4R\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is a point of maxima.\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>hR\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4R\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>R-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4R\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>8R\u003c/mn>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6R\u003c/mn>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8R\u003c/mn>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, Volume of Cone, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8R\u003c/mn>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4R\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>πR\u003c/mi>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Volume of sphere.\u003c/p>","order":98,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-96-prove-that-the-volume-of-the-largest-cone-that-can-be-inscribed-in-a-sphere-lt-strong-qna-11427238"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398560,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427284,"answerId":11427290,"questionText":"\u003cp>\u003cstrong>98. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> \sqrt2.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r, h, l and Ø be the radius, height, slant height and semi-vertical angle respectively of the cone. i.e., r, h, l>0.\u003c/p>\u003cp>Then, Volume V of the cone is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cbr>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>l\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>l\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>dV\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>l\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2V\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734083051phpEK0yM8_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734083051phpEK0yM8.jpeg\" alt=\"\" width=\"348\" height=\"430\">\u003c/picture>\u003c/div>\u003c/div>","order":99,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-98-show-that-the-semi-vertical-angle-of-the-cone-of-the-maximum-volume-and-of-given-slant-qna-11427284"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398564,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427350,"answerId":11427364,"questionText":"\u003cp>\u003cstrong>100. The point on the curve x2 = 2y which is nearest to the point (0, 5) is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> (2\sqrt2 \u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> ,4) \u003cstrong>(B)\u003c/strong> (2\sqrt2 \u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> ,0) \u003cstrong>(C)\u003c/strong> (0, 0) \u003cstrong>(D)\u003c/strong> (2, 2)\u003c/p>","answerText":"\u003cp>The equation of the given curve is\u003c/p>\u003cp>\u003cem>x\u003csup>2\u003c/sup>\u003c/em> = 2y.\u003c/p>\u003cp>Let p (x, y) be a point on the curve.\u003c/p>\u003cp>The distance of p (x, y) from (0, 5) is say S is given by\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734083902phphfzVA2_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734083902phphfzVA2.jpeg\" alt=\"\" width=\"223\" height=\"37\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Let z = \u003cem>s\u003c/em>\u003csup>2\u003c/sup> = \u003cem>x\u003c/em>\u003csup>2 \u003c/sup>+ \u003cem>y\u003c/em>\u003csup>2\u003c/sup> + 25 - 10\u003cem>y\u003c/em> = 2\u003cem>y\u003c/em> + \u003cem>y\u003c/em>\u003csup>2 \u003c/sup>-10\u003cem>y\u003c/em> + 25\u003c/p>\u003cp>z = \u003cem>y\u003c/em>\u003csup>2\u003c/sup> - 8\u003cem>y\u003c/em> + 25\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At y = 4, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>y = 4, is point of minimum distance.\u003c/p>\u003cp>So, \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 2\u003cem>y\u003c/em>->\u003cem>x\u003c/em>2 = 2 \times 4-> \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = 8 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the point of the nearest distance are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>2\sqrt2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\sqrt2\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Option (A) is correct.\u003c/p>","order":100,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-100-the-point-on-the-curve-x2-2y-which-is-nearest-to-the-point-0-5-is-lt-strong-gt-lt-p-gt-qna-11427350"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398566,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427378,"answerId":11427382,"questionText":"\u003cp>\u003cstrong>101. For all real values of x, the minimum value of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> 0 \u003cstrong>(B)\u003c/strong> 1 \u003cstrong>(C)\u003c/strong> 3 \u003cstrong>(D)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","answerText":"\u003cp>We have,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmenclose notation=\"updiagonalstrike\">\u003cmi>f\u003c/mi>\u003c/menclose>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At f(x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = 1 and x = -1.\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x = -1, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The maximum value of f(x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, option (D) is correct.\u003c/p>","order":101,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-101-for-all-real-values-of-x-the-minimum-value-of-lt-span-title-quot-click-to-copy-mathml-qna-11427378"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398568,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427390,"answerId":11427394,"questionText":"\u003cp>\u003cstrong>102. The maximum value of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> , 0<x<1 is<=\"\" strong=\"\"></x<1>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> \u003cstrong>(B)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cstrong>(C)\u003c/strong> 1 \u003cstrong>(D)\u003c/strong> 0\u003c/p>","answerText":"\u003cp>We have,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>X(x - 1)\u003c/mi>\u003cmi>+\u003c/mi>\u003c/math>\u003cbr>\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo>\u003c/mo>\u003cmn>1]\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At f(x) = 0.\u003c/p>\u003cp>2x - 1 = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>0(0 - 1)\u003c/mn>\u003c/math>\u003cbr>\u003cmath>\u003cmrow>\u003cmo>\u003c/mo>\u003cmn>+\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>(1 - 1)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>1]\u003c/mn>\u003c/math>\u003cbr>\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Option (B) is\u003c/p>\u003cp>Hence maximum value of f(x) = at x = 1 and x = 0.\u003c/p>\u003cp>Option (c) is correct.\u003c/p>","order":102,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-102-the-maximum-value-of-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-qna-11427390"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398570,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427406,"answerId":11427416,"questionText":"\u003cp>\u003cstrong>103. Using differentials, find the approximate value of each of the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(a) \u003c/strong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","answerText":"\u003cp>\u003cstrong>(a) \u003c/strong>Consider \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmi>L\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>&\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>T\u003c/mi>\u003cmi>h\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> is approximately equal to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> and is given by,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the approximate value of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>6\u003c/mn>\u003cmn>6\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=0.677\u003c/p>\u003cp>\u003cstrong>(b)\u003c/strong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(b) Consider \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmi>L\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmo>&\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734087182phpQyBEuv_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734087182phpQyBEuv.jpeg\" alt=\"\" width=\"541\" height=\"148\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> is approximately equal to \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> and is given by,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the approximate value of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>33\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003cbr>\u003c/span>is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>9\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":103,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-103-using-differentials-find-the-approximate-value-of-each-of-the-following-lt-strong-gt-qna-11427406"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398572,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427426,"answerId":11427432,"questionText":"\u003cp>\u003cstrong>104. Show that the function given by \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> has maximum at x = e.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f(x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmunder accentunder=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo stretchy=\"true\">_\u003c/mo>\u003c/munder>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At extreme points, f(x) = 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>e\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x = e, f\"(e) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>l\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>g\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = e is a point of maximum.\u003c/p>","order":104,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-104-show-that-the-function-given-by-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-qna-11427426"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398574,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427444,"answerId":11427454,"questionText":"\u003cp>\u003cstrong>105. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘b’ and ‘x’ be the fixed base and equal side of isosceales triangle.\u003c/p>\u003cp>Then, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> cm/s (Ø decreasing).\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734087295php6A19As_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734087295php6A19As.jpeg\" alt=\"\" width=\"385\" height=\"479\">\u003c/picture>\u003c/div>\u003c/div>","order":105,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-105-the-two-equal-sides-of-an-isosceles-triangle-with-fixed-base-b-are-decreasing-at-the-qna-11427444"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398576,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427460,"answerId":11427470,"questionText":"\u003cp>\u003cstrong>106. Find the equation of the normal to curve x\u003csup>2 \u003c/sup>= 4y which passes through the point (1, 2)\u003c/strong>\u003c/p>","answerText":"\u003cp>Equation of the curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, the slope of the normal at point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Equation of the normal at \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>","order":106,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-106-find-the-equation-of-the-normal-to-curve-x-lt-sup-gt-2-lt-sup-gt-4y-which-passes-through-qna-11427460"},"anaFaqSource":"manual","generationSource":"manual","templateId":163,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398578,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427478,"answerId":11427488,"questionText":"\u003cp>\u003cstrong>107. Show that the normal at any point θ to the curve x = a cosθ + a θ sin θ, y = a sinθ - aθ cosθ is at a constant distance from the origin.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Slope of the normal at any point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The equation of the normal at a given point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is given by,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, the perpendicular distance of the normal from the origin is\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734087410phpZY1Fs7_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734087410phpZY1Fs7.jpeg\" alt=\"\" width=\"396\" height=\"70\">\u003c/picture>\u003c/div>\u003c/div>","order":107,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-107-show-that-the-normal-at-any-point-theta-to-the-curve-x-a-cos-theta-a-theta-sin-theta-qna-11427478"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398580,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427496,"answerId":11427502,"questionText":"\u003cdiv>\u003cstrong>108. Find the intervals in which the function f given by\u003c/strong>\u003c/div>\n\u003cdiv>\n\u003cp>\u003cstrong>f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is (i) increasing (ii) decreasing\u003c/strong>\u003c/p>\n\u003c/div>","answerText":"\u003cp>We have, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> as cos x lies in [1, 1].\u003c/p>\u003cp>So, (i) for increasing, f(x) \geq 0.\u003c/p>\u003cp>cosx \geq 0.\u003c/p>\u003cp>x lies in Ist and IVth quadrant.\u003c/p>\u003cp>i.e., f(x) is increasing for \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>\leq\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\leq\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(ii) for decreasing, f(x) \leq 0.\u003c/p>\u003cp>cosx \leq 0.\u003c/p>\u003cp>x lies in IInd and IIIrd quadrant.\u003c/p>\u003cp>i.e., f(x) is decreasing for \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\leq\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\leq\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>","order":108,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-div-gt-lt-strong-gt-108-find-the-intervals-in-which-the-function-f-given-by-lt-strong-gt-lt-div-gt-lt-div-gt-qna-11427496"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398582,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427522,"answerId":11427526,"questionText":"\u003cp>\u003cstrong>109. Find the intervals in which the function f given by \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i)\u003c/strong> Increasing \u003cstrong>(ii)\u003c/strong> Decreasing\u003c/p>","answerText":"\u003cp>We have, f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\neq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> { \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo> \u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\neq\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So we have three disjoint internal i.e.,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>\infty\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>,\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>\infty\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>\infty\u003c/mi>\u003cmo>,\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo> \u003c/mo>\u003cmtext> and\u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmo> \u003c/mo>\u003cmtext> at\u003c/mtext>\u003cmo> \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmo> \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f(x) is increasing.\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo> \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>-\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo> \u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo> at\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1and \u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f(x) is decreasing.\u003c/p>\u003cp>When \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>\infty\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo> \u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo> \u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo> \u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> on\u003c/mtext>\u003cmo> \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo> at\u003c/mo>\u003cmo> \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f(x) is increasing.\u003c/p>\u003cp>f(x) is increasing for x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>(\infty,1) and [1, \infty] and decreasing for x\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\in\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>[1, 1].\u003c/p>","order":109,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-109-find-the-intervals-in-which-the-function-f-given-by-lt-span-title-quot-click-to-copy-qna-11427522"},"anaFaqSource":"manual","generationSource":"manual","templateId":101,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398584,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427544,"answerId":11427554,"questionText":"\u003cp>\u003cstrong>110. Find the maximum area of an isosceles triangle inscribed in the ellipse\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/math>\u003c/span> with its vertex at one end of the major axis.\u003c/strong>\u003c/p>","answerText":"\u003cp>The given equation of the ellipse is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/math>\u003c/span> (1)\u003c/p>\u003cp>Let the major axis be along x-axis so, vertex is at \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>, \u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Let ΔABC be the isosceles triangle inscribed on the\u003c/p>\u003cp>ellipse with one vertex C at (a, 0).\u003c/p>\u003cp>Then, let A have Co-ordinate (x0, yo) from figure.\u003c/p>\u003cp>So, Co-ordinate of B = (x0, y0)\u003c/p>\u003cp>As A and B lies on the ellipse, from equation (i), \u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734088430phpEdYX3B_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734088430phpEdYX3B.jpeg\" alt=\"\" width=\"377\" height=\"290\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734088443phpfqyN4H_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734088443phpfqyN4H.jpeg\" alt=\"\" width=\"398\" height=\"473\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734088456phpW5DTG6_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734088456phpW5DTG6.jpeg\" alt=\"\" width=\"296\" height=\"386\">\u003c/picture>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734088456phpW5DTG6.jpeg\"1 media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734088443phpfqyN4H_480x360.jpeg\"4 alt=\"\" width=\"231\" height=\"284\">\u003c/picture>\u003c/div>\u003c/div>","order":110,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-110-find-the-maximum-area-of-an-isosceles-triangle-inscribed-in-the-ellipse-lt-strong-gt-qna-11427544"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398586,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427564,"answerId":11427572,"questionText":"\u003cp>\u003cstrong>111. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m\u003csup>3\u003c/sup>. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x and y meters be the length and breath of the rectangular base of the tank respectively.\u003c/p>\u003cp>Then, volume V of the tank is\u003c/p>\u003cp>V = length \times depth \times breath.\u003c/p>\u003cp>V = 2xy = 8m\u003csup>3\u003c/sup>(given).\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Let ‘t’ be the total cost of building the tank.\u003c/p>\u003cp>Then, t = cost of base + cost of sides.\u003c/p>\u003cp>= 70xy + 45[4x+4y] {there are four sides.\u003c/p>\u003cp>= 70xy + 180x+ 180y.\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>7\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>\cdot\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = \pm 2\u003c/p>\u003cp>x = 2, (x = length and it cannot be negative)\u003c/p>\u003cp>At x = 2, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = 2 is point of maxima.\u003c/p>\u003cp>Hence, minimum cost = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 280 + 360 + 360 = 1000.\u003c/p>","order":111,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-111-a-tank-with-rectangular-base-and-rectangular-sides-open-at-the-top-is-to-be-constructed-qna-11427564"},"anaFaqSource":"manual","generationSource":"manual","templateId":91,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398588,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427582,"answerId":11427588,"questionText":"\u003cp>\u003cstrong>112. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let r and s be the radius of the circle and length of side of square.\u003c/p>\u003cp>Then, sum of perimeter of circle and square = k\u003c/p>\u003cp>2πr +4s = k\u003c/p>\u003cp>s = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>\times\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>The area A be the total areas of the circle and square.\u003c/p>\u003cp>Then, A = π\u003cem>r\u003c/em>\u003csup>2\u003c/sup> + \u003cem>s\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>A\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>\cdot\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2A\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>A\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>k\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>k\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2A\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734088717php8O6Esh_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734088717php8O6Esh.jpeg\" alt=\"\" width=\"239\" height=\"158\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>s = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>s = 2r.\u003c/p>\u003cp>Hence, proved.\u003c/p>","order":112,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-112-the-sum-of-the-perimeter-of-a-circle-and-square-is-k-where-k-is-some-constant-prove-that-qna-11427582"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398590,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427608,"answerId":11427616,"questionText":"\u003cp>\u003cstrong>113. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.\u003c/strong>\u003c/p>","answerText":"\u003cp>Let ‘x’ metre be the radius of the semi-circular opening mounded on the length ‘2x’ side of rectangle. Then, let ‘y’ be the breadth of the rectangle.\u003c/p>\u003cp>Then, perimeter of the window = 10m\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x + 2x + 2y + = 10.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Let the area of the window be A.\u003c/p>\u003cp>Then, A = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> [-π\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 4\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 20\u003cem>x\u003c/em>].\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> [ -2πx - 8x + 20]\u003c/p>\u003cp>And \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmover>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mover>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> [ -2π - 8] = -π -4 = -( π+ 4)\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> [ -2πx - 8x + 20] = 0\u003c/p>\u003cp>2x + 8x = 20\u003c/p>\u003cp>x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2A\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = -( π+ 4) < 0\u003c/p>\u003cp>Øx = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is a point of minima.\u003c/p>\u003cp>And y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>\times\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Ø Dimensions of the window are\u003c/p>\u003cp>length = 2x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>breadth = y \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>m\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>radius = y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>m\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":113,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-113-a-window-is-in-the-form-of-a-rectangle-surmounted-by-a-semicircular-opening-the-total-qna-11427608"},"anaFaqSource":"manual","generationSource":"manual","templateId":67,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398592,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427632,"answerId":11427644,"questionText":"\u003cp>\u003cstrong>114. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>Let P be the point on hypotenuse of a triangle. ABC, t angle at B.\u003c/p>\u003cp>Which is at distance a& b from the sides of the triangle.\u003c/p>\u003cp>Let < BAC = < MPC = Ø.\u003c/p>\u003cp>Then, in\dots ΔANP,\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734089111phpN0pSvT_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734089111phpN0pSvT.jpeg\" alt=\"\" width=\"324\" height=\"449\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>θ\u003c/mi>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At tan Ø = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmover accent=\"true\">\u003cmi>q\u003c/mi>\u003cmo>?\u003c/mo>\u003c/mover>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>+\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>v\u003c/mi>\u003cmi>e\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>.\u003c/mo>\u003cmo>>\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> { Øall trigonometric f\u003cem>x\u003c/em>\u003csup>n\u003c/sup> are + ve in I\u003csup>st\u003c/sup> quadrant}.\u003c/p>\u003cp>So, z is least for tanØ = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As, Sec\u003csup>2\u003c/sup>Ø = 1 + tan\u003csup>2\u003c/sup>Ø = 1 + \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>secØ = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And tan\u003csup>2\u003c/sup>Ø = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>t\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>θ\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>cot\u003csup>2\u003c/sup>Ø = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And cosec\u003csup>2\u003c/sup>Ø = 1 + cot\u003csup>2\u003c/sup>Ø = 1 + \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>cosecØ = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734089330php4Ht7A8.jpeg\" alt=\"\" width=\"383\" height=\"128\">\u003c/picture>\u003c/div>\u003c/div>","order":114,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-114-a-point-on-the-hypotenuse-of-a-triangle-is-at-distance-a-and-b-from-the-sides-of-the-qna-11427632"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398594,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427650,"answerId":11427658,"questionText":"\u003cp>\u003cstrong>115. Find the points at which the function f given by f (x) = (x - 2)\u003csup>4\u003c/sup> (x + 1)\u003csup>3\u003c/sup> has (i) local maxima (ii) local minima (iii) point of inflexion\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, \u003c/p>\u003cp>f (x) = (\u003cem>x\u003c/em>- 2)\u003csup>4\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup>.\u003c/p>\u003cp>So, f (x) = (\u003cem>x\u003c/em>- 2)\u003csup>4\u003c/sup>. 3 (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> + (\u003cem>x\u003c/em> + 1)\u003csup>3\u003c/sup>. 4 (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup>.\u003c/p>\u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> [\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> [3 (\u003cem>x\u003c/em>- 2) + 4 (\u003cem>x\u003c/em> + 1)]\u003c/p>\u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (3\u003cem>x\u003c/em>- 6 + 4\u003cem>x\u003c/em> + 4)\u003c/p>\u003cp>= (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup> (7\u003cem>x\u003c/em>- 2).\u003c/p>\u003cp>At f (x) = 0.\u003c/p>\u003cp> (\u003cem>x\u003c/em>- 2)\u003csup>3\u003c/sup> (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>. (7\u003cem>x\u003c/em>- 2) = 0.\u003c/p>\u003cp>x = 2, x = -1 or x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As (\u003cem>x\u003c/em> + 1)\u003csup>2\u003c/sup>> 0, we shave evaluate for the remaining factor.\u003c/p>\u003cp>At x = 2, \u003c/p>\u003cp>When x< 2, f (x) = ( -ve) (+ ve) (+ ve) = ( -ve) < 0.\u003c/p>\u003cp>When x> 2, f (x) = (+ ve) (+ ve) (+ ve) = (+ ve) > 0.\u003c/p>\u003cp>Øf (x) change from ( -ve) to (+ ve) as x increases\u003c/p>\u003cp>So, x = 2 is a point of local minima\u003c/p>\u003cp>At x = -1.\u003c/p>\u003cp>When x< -1, f (x) = ( -ve) (+ ve) ( -ve) =\notin, ve > 0.\u003c/p>\u003cp>When x> -1, f (x) = ( -ve) (+ ve) (+ ve) =\notin, ve > 0.\u003c/p>\u003cp>So, f (x) does not change through x -1.\u003c/p>\u003cp>Hence, x = -1 is a point of infixion\u003c/p>\u003cp>At x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>When x< \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) = ( -ve) (+ ve) ( -ve) = ( -ve > 0.\u003c/p>\u003cp>When x< \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) = ( -ve) (+ ve) (+ ve) = ( -ve < 0.\u003c/p>\u003cp>f (x) change from (+ ve) to ( -ve) as x increases through \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> is a point of local maxima\u003c/p>","order":115,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-115-find-the-points-at-which-the-function-f-given-by-f-x-x-ndash-2-lt-sup-gt-4-lt-sup-gt-qna-11427650"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398596,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427674,"answerId":11427680,"questionText":"\u003cp>\u003cstrong>116. Find the absolute maximum and minimum values of the function f given by f (x) =cos\u003csup>2\u003c/sup> x + sin x, x \in [0, π]\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, \u003c/p>\u003cp>f (x) = cos\u003csup>2\u003c/sup>x + sin x, x\in [0, π ].\u003c/p>\u003cp>So, f (x) = 2 cos x ( -sin x) + cos x = cos x (1 - 2 sin x).\u003c/p>\u003cp>At f (x) = 0\u003c/p>\u003cp>cosx (1 - 2 sin x) = 0\u003c/p>\u003cp>cosx = 0 or 1 - 2 sin x = 0\u003c/p>\u003cp>cosx = cos \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> or sin x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = sin \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = sin \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>x= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> , x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> and x = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> [0, π ].\u003c/p>\u003cp>So, f \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> = cos\u003csup>2\u003c/sup>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> + sin \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> = 1.\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734089825phpXz99UG_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734089825phpXz99UG.jpeg\" alt=\"\" width=\"308\" height=\"145\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>Absolute minimum of f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> and absolute minimum of f (x) = 1.\u003c/p>","order":116,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-116-find-the-absolute-maximum-and-minimum-values-of-the-function-f-given-by-f-x-cos-lt-sup-qna-11427674"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398598,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427686,"answerId":11427696,"questionText":"\u003cp>\u003cstrong>117. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x be the radius of the sphere ad x be radius of the right circular cone.\u003c/p>\u003cp>Let height of cone = y\u003c/p>\u003cp>Then, in ΔOBA,\u003c/p>\u003cp>(\u003cem>y\u003c/em>-\u003cem>r\u003c/em>)\u003csup>2\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = \u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cem>y\u003c/em>\u003csup>2\u003c/sup> + \u003cem>r\u003c/em>\u003csup>2\u003c/sup>- 2\u003cem>ry\u003c/em> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> = \u003cem>r\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cem>x\u003c/em>\u003csup>2\u003c/sup>=2\u003cem>ry \u003c/em>-\u003cem> y\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>So, the volume V of the cone is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>V=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmo>\cdot\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\cdot\u003c/mo>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>r\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>u\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\timeshight\u003c/mo>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\times\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>And \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>y\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>V\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>4\u003cem>x \u003c/em>-\u003cem>y\u003c/em>- 3\u003cem>y\u003c/em>\u003csup>2\u003c/sup> = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> asy> 0.\u003c/p>\u003cp>At y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>v\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Ø V is maximum when y = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":117,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-117-show-that-the-altitude-of-the-right-circular-cone-of-maximum-volume-that-can-be-inscribed-qna-11427686"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398600,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427704,"answerId":11427710,"questionText":"\u003cp>\u003cstrong>118. Let f be a function defined on [a, b] such that f '(x) > 0, for all x \in (a, b).\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Then prove that f is an increasing function on (a, b).\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, \u003c/p>\u003cp>f (x) defined on [a, b]\u003c/p>\u003cp>And f (x) > 0 \forall x \in [a, b].\u003c/p>\u003cp>Let \u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2 \u003c/sub>\in [a, b] and \u003cem>x\u003c/em>\u003csub>2\u003c/sub>>\u003cem>x\u003c/em>\u003csub>1\u003c/sub>\u003c/p>\u003cp>In the internal \u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2\u003c/sub>], f (x) will also be continuous and differentiable.\u003c/p>\u003cp>Hence by mean value theorem, there exist c [\u003cem>x\u003c/em>\u003csub>1\u003c/sub>, \u003cem>x\u003c/em>\u003csub>2\u003c/sub>] such that\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f (x) > 0 \forall x \in [a, b].\u003c/p>\u003cp>Then, f (c) > 0.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>i.e., f (\u003cem>x\u003c/em>\u003csub>2\u003c/sub>) -\u003cem>f\u003c/em> (\u003cem>x\u003c/em>\u003csub>1\u003c/sub>) > 0\u003c/p>\u003cp>f (\u003cem>x\u003c/em>\u003csub>2\u003c/sub>) >\u003cem>f\u003c/em> (\u003cem>x\u003c/em>\u003csub>1\u003c/sub>).\u003c/p>\u003cp>Hence, the function f (x) is always increasing on [a, b]\u003c/p>","order":118,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-118-let-f-be-a-function-defined-on-such-that-f-prime-x-amp-gt-0-for-all-x-isin-a-b-lt-strong-qna-11427704"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398602,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427716,"answerId":11427726,"questionText":"\u003cp>\u003cstrong>119. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/\sqrt3. Also find the maximum volume.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cspan data-teams=\"true\">Kindly go through the solution\u003c/span>\u003c/p>\u003cdiv class=\"photo-widget-full\">\u003cdiv class=\"figure\">\u003cpicture>\u003csource srcset=\"\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1752129315phpKvzvuq.jpeg\" alt=\"\" width=\"367\" height=\"526\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cspan data-teams=\"true\">\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734093061phpcHOQmp.jpeg\" alt=\"\" width=\"292\" height=\"126\">\u003c/picture>\u003c/span>\u003c/p>","order":119,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-119-show-that-the-height-of-the-cylinder-of-maximum-volume-that-can-be-inscribed-in-a-sphere-qna-11427716"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398606,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427748,"answerId":11427754,"questionText":"\u003cp>\u003cstrong>121. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h\u003c/strong>\u003c/p>","answerText":"\u003cp>Let x be the depth of the wheat inside the cylindrical tank is with radius = 10 cm\u003c/p>\u003cp>Then, Volume V of the cylindrical tank is\u003c/p>\u003cp>V = π (10)\u003csup>2\u003c/sup>\u003cem>x\u003c/em> = 100πp\u003cem>x\u003c/em> m\u003csup>3\u003c/sup>\u003c/p>\u003cp>As \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>t\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>π\u003c/mi>\u003cmn>3\u003c/mn>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>1\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> i e, rate of increasing of depth\u003c/p>\u003cp>option (A) is correct\u003c/p>","order":120,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-121-a-cylindrical-tank-of-radius-10-m-is-being-filled-with-wheat-at-the-rate-of-314-cubic-qna-11427748"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398612,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427794,"answerId":11427798,"questionText":"\u003cp>\u003cstrong>124. The normal to the curve x\u003csup>2\u003c/sup> = 4y passing (1,2) is (A) x + y = 3 (B) x - y = 3 (C) x + y = 1 (D) x - y = 1\u003c/strong>\u003c/p>","answerText":"\u003cp>The equation of the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Differentiating with respect to x, we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>The slope of the normal to the given curve at point (h,k) is given by,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> Equation of the normal at point (h,k) is given as:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Now, it is given that the normal passes through the point (1,2).\u003c/p>\u003cp>Therefore, we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>o\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Since (h,k) lies on the curve \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> ,we have \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>k\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>From equation (i), we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>-\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>k\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the equation of the normal is given as:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Therefore, option (A) is correct.\u003c/p>","order":121,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-124-the-normal-to-the-curve-x-lt-sup-gt-2-lt-sup-gt-4y-passing-1-2-is-a-x-y-3-b-x-ndash-y-qna-11427794"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398614,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11427810,"answerId":11427814,"questionText":"\u003cp>\u003cstrong>125. The points on the curve 9y\u003csup>2\u003c/sup> = x\u003csup>3\u003c/sup> , where the normal to the curve makes equal intercepts with the axes are\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(A) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> (B) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (C) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (D) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\pm\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>The equation of the given curve is \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Differentiate with respect to x, we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmi>y\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>The slope of the normal to the given curve at point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>∴\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> The equation of the normal to the curve at \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> is\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>y\u003c/mi>\u003cmo>-\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>y\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>It is given that the normal makes intercepts with the axes.\u003c/p>\u003cp>Therefore, we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>∴\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Also, the point \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>,\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> lies on the curve, so we have\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo>.\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>i\u003c/mi>\u003cmi>i\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>From (i) and (ii), we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\Rightarrow\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>\Rightarrow\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>From (ii), we have:\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmrow>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>y\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmo>=\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, the required points are \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmo>\pm\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> .\u003c/p>\u003cp>Therefore, option (A) is correct.\u003c/p>","order":122,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-125-the-points-on-the-curve-9y-lt-sup-gt-2-lt-sup-gt-x-lt-sup-gt-3-lt-sup-gt-where-the-normal-qna-11427810"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398436,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11425192,"answerId":11425196,"questionText":"\u003cp>\u003cstrong>36. Prove that the function given by f (x) = x\u003csup>3\u003c/sup> - 3x\u003csup>2\u003c/sup> + 3x - 100 is increasing in R.\u003c/strong>\u003c/p>","answerText":"\u003cp>We have, f (x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup>- 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> 3\u003cem>x\u003c/em>- 100.\u003c/p>\u003cp>So, f (x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 6\u003cem>x\u003c/em> + 3 = 3 (\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 2\u003cem>x\u003c/em> + 1) = 3 (\u003cem>x\u003c/em>- 1)\u003csup>2\u003c/sup>\u003c/p>\u003cp>For \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp> (\u003cem>x\u003c/em>- 10)\u003csup>2\u003c/sup>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> , 0 for x = 1.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 3 (\u003cem>x\u003c/em>- 1)\u003csup>2\u003c/sup> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\geq\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>∴f (x) is increasing on \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>? \u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":123,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-36-prove-that-the-function-given-by-f-x-x-lt-sup-gt-3-lt-sup-gt-ndash-3x-lt-sup-gt-2-lt-sup-qna-11425192"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":41398518,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180144","questionId":11426882,"answerId":11426886,"questionText":"\u003cp>\u003cstrong>77. Prove that the following functions do not have maxima or minima:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i)\u003c/strong> f(x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>\u003c/p>","answerText":"\u003cp>\u003cstrong> (i) \u003c/strong>We have, f (x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> f (x) = \u003cem>e\u003csup>x\u003c/sup>\u003c/em>.\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>'\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>e\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At, extreme points, \u003c/p>\u003cp>f (x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> \u003cem>e\u003csup>x\u003c/sup>\u003c/em> = 0 which has no real ‘a’ value\u003c/p>\u003cp>∴f (x) has with maximum or minima\u003c/p>\u003cp>\u003cstrong> (ii)\u003c/strong> g (x) = log x\u003c/p>\u003cp>A (ii)\u003c/p>\u003cp>We have, g (x) = log x, \u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> g (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At extreme points, \u003c/p>\u003cp>g (x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\to\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 1 = 0 which is not true.\u003c/p>\u003cp>∴g (x) was value minima or maxima\u003c/p>\u003cp>\u003cstrong> (iii)\u003c/strong> h (x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>x\u003c/em> + 1.\u003c/p>\u003cp>A (iii)\u003c/p>\u003cp>We have, h (x) = \u003cem>x\u003c/em>\u003csup>3\u003c/sup> + \u003cem>x\u003c/em>\u003csup>2\u003c/sup> + \u003cem>x\u003c/em> + 1.\u003c/p>\u003cp>h (x) = 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2x + 1\u003c/p>\u003cp>At extreme points, \u003c/p>\u003cp>h (x) = 0\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 3\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 2x + 1 = 0\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1734072775phpqEP6tT_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1734072775phpqEP6tT.jpeg\" alt=\"\" width=\"211\" height=\"161\">\u003c/picture>\u003c/div>\u003c/div>","order":124,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-77-prove-that-the-following-functions-do-not-have-maxima-or-minima-lt-strong-gt-lt-p-gt-lt-qna-11426882"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"isAIGenerated":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false}]}},"prepAnaData":{"fetchTypeToFaqData":{"180458":[{"anaFaqId":53135550,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180458","questionId":11692028,"answerId":11692060,"questionText":"\u003cp>\u003cstrong>The sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-sum-of-absolute-maximum-and-absolute-minimum-values-of-the-function-f-x-2x-lt-sup-gt-qna-11692028"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7756,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>/\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7758,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":true},{"id":7760,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7762,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135552,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180458","questionId":11687542,"answerId":11687574,"questionText":"\u003cp>\u003cstrong>Let f : R® R be a continuous function satisfying f(x) + f(x + k) = n, for all x \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\in\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> R where k > 0 and n is a positive integer. If \u003c/strong>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmi>t\u003c/mi> \u003cmi>h\u003c/mi> \u003cmi>e\u003c/mi> \u003cmi>n\u003c/mi> \u003cmo>:\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003cstrong> \u003c/strong>\u003c/p>","answerText":"\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>\forall\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>\in\u003c/mo> \u003cmi>R\u003c/mi> \u003cmo>,\u003c/mo> \u003cmtext> \u003c/mtext> \u003cmtext> \u003c/mtext> \u003cmo>&\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>Replace x by x + k. \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmi>n\u003c/mi> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>From (i) & (ii), f(x + 2k) = f(x).\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  is periodic with period = 2k.\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>n\u003c/mi> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmo>.\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>i\u003c/mi> \u003cmi>i\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsub> \u003cmrow> \u003cmi>I\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmtext> \u003c/mtext> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>put x = t + k\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>k\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>n\u003c/mi> \u003cmstyle displaystyle=\"true\"> \u003cmrow> \u003cmunderover> \u003cmo>\int\u003c/mo> \u003cmrow> \u003cmn>0\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>k\u003c/mi> \u003c/mrow> \u003c/munderover> \u003cmrow> \u003cmi>n\u003c/mi> \u003cmi>d\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/mrow> \u003c/mstyle> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>n\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmi>k\u003c/mi> \u003cmo>.\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>  \u003c/p>\u003cp>  \u003c/p>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-let-f-r-reg-r-be-a-continuous-function-satisfying-f-x-f-x-k-n-for-all-x-lt-strong-gt-lt-span-qna-11687542"},"anaFaqSource":"manual","generationSource":"manual","templateId":141,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7204,"optionText":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 4nk\u003c/p>","order":0,"correct":false},{"id":7206,"optionText":"\u003cp>I\u003csub>1\u003c/sub> + 2I\u003csub>2\u003c/sub> = 2nk\u003c/p>","order":1,"correct":false},{"id":7208,"optionText":"\u003cp> I\u003csub>1\u003c/sub>+ nI\u003csub>2\u003c/sub> = 4n\u003csup>2\u003c/sup>k\u003c/p>","order":2,"correct":true},{"id":7210,"optionText":"\u003cp> I\u003csub>1\u003c/sub> + nI\u003csub>2\u003c/sub> = 6n\u003csup>2\u003c/sup>k\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135554,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180458","questionId":11638756,"answerId":11638760,"questionText":"\u003cp>\u003cstrong>The number of points, where the function f : R R,\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math> is NOT differentiable, is:\u003c/strong>\u003c/p>","answerText":"\u003cp> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>x = 1, 4 (doubtful points)\u003c/p>\u003cp>Diff. at x = 1\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>-\u003c/mo>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>-\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>R\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>L\u003c/mi>\u003cmi>H\u003c/mi>\u003cmi>D\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>\to\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>-\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>-\u003c/mo>\u003cmo>\infty\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmo>}\u003c/mo>\u003c/mrow>\u003cmo>\Rightarrow\u003c/mo>\u003c/mrow>\u003c/math> Not diff. at x = 4\u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-number-of-points-where-the-function-f-r-r-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-lt-qna-11638756"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":1894,"optionText":"\u003cp>1\u003c/p>","order":0,"correct":false},{"id":1896,"optionText":"\u003cp>2\u003c/p>","order":1,"correct":true},{"id":1898,"optionText":"\u003cp>3\u003c/p>","order":2,"correct":false},{"id":1900,"optionText":"\u003cp>4\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true}],"180460":[{"anaFaqId":49226530,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11691928,"answerId":11691952,"questionText":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","answerText":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-for-the-function-f-x-4-log-lt-sub-gt-e-lt-sub-gt-x-ndash-1-ndash-2x-lt-sup-gt-2-lt-sup-gt-qna-11691928"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7740,"optionText":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","order":0,"correct":false},{"id":7742,"optionText":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","order":1,"correct":false},{"id":7744,"optionText":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","order":2,"correct":true},{"id":7746,"optionText":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":49226532,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11691968,"answerId":11692008,"questionText":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-if-the-tangent-at-the-point-x-lt-sub-gt-1-lt-sub-gt-y-lt-sub-gt-1-lt-sub-gt-on-the-curve-qna-11691968"},"anaFaqSource":"manual","generationSource":"manual","templateId":123,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7748,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7750,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":false},{"id":7752,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7754,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":true}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":49226534,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"180460","questionId":11692028,"answerId":11692060,"questionText":"\u003cp>\u003cstrong>The sum of absolute maximum and absolute minimum values of the function f(x) = |2x\u003csup>2\u003c/sup> + 3x + 2| + sin x cos x in the interval [0, 1] is :\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>?\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>?\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>?\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>?\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-sum-of-absolute-maximum-and-absolute-minimum-values-of-the-function-f-x-2x-lt-sup-gt-qna-11692028"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7756,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>/\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7758,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":true},{"id":7760,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>5\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7762,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true}],"210580":[{"anaFaqId":53135544,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210580","questionId":11691674,"answerId":11691686,"questionText":"\u003cp>\u003cstrong>The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :\u003c/strong>\u003c/p>","answerText":"\u003cp>Surface area, S = 4pr\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmo>.\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>r\u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003cmi>? \u003c/mi> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>t\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>a\u003c/mi> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814779295\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>s\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>d\u003c/mi> \u003cmi>t\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmo>? \u003c/mo> \u003cmi>s\u003c/mi> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>? \u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>? \u003c/mi> \u003cmsup> \u003cmrow> \u003cmi>r\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmi>k\u003c/mi> \u003cmi>t\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814779296\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>Initially t = 0, r = 3\u003c/p>\u003cp>c = 36 p\u003c/p>\u003cp>When t = 5, r = 7, k = 32p\u003c/p>\u003cp>When t = 9, r = r, r = 9\u003c/p>\u003cp> \u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-the-surface-area-of-a-balloon-of-spherical-shape-being-inflated-increases-at-a-constant-rate-qna-11691674"},"anaFaqSource":"manual","generationSource":"manual","templateId":69,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7684,"optionText":"\u003cp>9\u003c/p>","order":0,"correct":true},{"id":7686,"optionText":"\u003cp>10\u003c/p>","order":1,"correct":false},{"id":7688,"optionText":"\u003cp>11\u003c/p>","order":2,"correct":false},{"id":7690,"optionText":"\u003cp>12\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135546,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210580","questionId":11691928,"answerId":11691952,"questionText":"\u003cp>\u003cstrong>For the function f(x) = 4 log\u003csub>e\u003c/sub>(x - 1) - 2x\u003csup>2\u003c/sup> + 4x + 5, x > 1, which one of the following is NOT correct?\u003c/strong>\u003c/p>","answerText":"\u003cp>f (x) = 4log\u003csub>e\u003c/sub> (x - 1) -2x\u003csup>2\u003c/sup> + 4x + 5, x > 1\u003c/p>\u003cp> \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>A\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>\Rightarrow\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>\Rightarrow\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmo>-\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1753254428phpzchmPq.jpeg\" alt=\"\" width=\"430\" height=\"69\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>option (A) is correct.\u003c/p>\u003cp> (B) f (x) = -1, has two solution\u003c/p>\u003cp>option (B) is correct\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>C\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmo>-\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1025\" DrawAspect=\"Content\" ObjectID=\"_1814781079\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmi>e\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>-\u003c/mo> \u003cmi>f\u003c/mi> \u003cmo>\"\u003c/mo> \u003cmo stretchy=\"false\"> (\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo stretchy=\"false\">)\u003c/mo> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>4\u003c/mn> \u003cmi>e\u003c/mi> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmo>-\u003c/mo> \u003cmi>e\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmi>e\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmn>8\u003c/mn> \u003cmo>>\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span> <!- [if gte mso 9]><xml> \u003co:OLEObject Type=\"Embed\" ProgID=\"Equation.DSMT4\" ShapeID=\"_x0000_i1026\" DrawAspect=\"Content\" ObjectID=\"_1814781080\"> \u003c/o:OLEObject></xml><! [endif]-> \u003c/p>\u003cp>option (C) is not correct\u003c/p>\u003cp> (D) f (e) = 4log\u003csub>e\u003c/sub> (e - 1) -2 (e\u003csup>2\u003c/sup> - 2e + 1) + 7 > 0\u003c/p>\u003cp>f (e + 1) = 4 - 2 (e + 1)\u003csup>2\u003c/sup> + 4 (e + 1) +5+ < 0\u003c/p>\u003cp>option (D) is correct \u003c/p>\u003c/div>\u003c/div>","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-for-the-function-f-x-4-log-lt-sub-gt-e-lt-sub-gt-x-ndash-1-ndash-2x-lt-sup-gt-2-lt-sup-gt-qna-11691928"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7740,"optionText":"\u003cp>f is increasing in (1, 2) and decreasing in (2, ¥)\u003c/p>","order":0,"correct":false},{"id":7742,"optionText":"\u003cp>f (x) = -1 has exactly two solutions\u003c/p>","order":1,"correct":false},{"id":7744,"optionText":"\u003cp>f’(e) - f’’ (2) < 0\u003c/p>","order":2,"correct":true},{"id":7746,"optionText":"\u003cp>f (x) = 0 has a root in the interval (e, e + 1)\u003c/p>","order":3,"correct":false}]},"experiment":false,"isAIGenerated":false,"crawlable":true},{"anaFaqId":53135548,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210580","questionId":11691968,"answerId":11692008,"questionText":"\u003cp>\u003cstrong>If the tangent at the point (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) on the curve y = x\u003csup>3\u003c/sup> + 3x\u003csup>2\u003c/sup> + 5 passes through the origin, then (x\u003csub>1\u003c/sub>, y\u003csub>1\u003c/sub>) does NOT lie on the curve:\u003c/strong>\u003c/p>","answerText":"\u003cp>f(x) =  \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>|\u003c/mo> \u003cmrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmo>|\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>2\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>2\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmi>x\u003c/mi> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>f\u003c/mi> \u003cmo>'\u003c/mo> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmrow> \u003cmo>{\u003c/mo> \u003cmrow> \u003cmtable> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmo>-\u003c/mo> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>-\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmn>0\u003c/mn> \u003cmo><\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mtd> \u003c/mtr> \u003cmtr columnalign=\"center\"> \u003cmtd columnalign=\"center\"> \u003cmn>4\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>2\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>,\u003c/mo> \u003c/mtd> \u003cmtd columnalign=\"center\"> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>\leq\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo><\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mtd> \u003c/mtr> \u003c/mtable> \u003c/mrow> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>  \u003c/p>\u003cp> f(1) = 3 + sin 1 cos 1 and \u003cspan contenteditable=\"false\"> \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>∴\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>+\u003c/mo> \u003cmi>f\u003c/mi> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmi>s\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>n\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>c\u003c/mi> \u003cmi>o\u003c/mi> \u003cmi>s\u003c/mi> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp> \u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-if-the-tangent-at-the-point-x-lt-sub-gt-1-lt-sub-gt-y-lt-sub-gt-1-lt-sub-gt-on-the-curve-qna-11691968"},"anaFaqSource":"manual","generationSource":"manual","templateId":123,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"MCQ","prepQuestionDetails":{"questionHint":"","questionDifficultyLevel":"HIGH","optionsDetails":[{"id":7748,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>8\u003c/mn> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":0,"correct":false},{"id":7750,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmn>9\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>8\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":1,"correct":false},{"id":7752,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>y\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>4\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":2,"correct":false},{"id":7754,"optionText":"\u003cp>\u003cspan class=\"mathml\" contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>-\u003c/mo> \u003cmsup> \u003cmrow> \u003cmi>y\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>","order":3,"correct":true}]},"experiment":false,"isAIGenerated":false,"crawlable":true}],"210582":[{"anaFaqId":53135542,"entityId":4459,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"210582","questionId":11638812,"answerId":11638818,"questionText":"\u003cp>\u003cstrong>Let f(x) = \u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>-\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>\in\u003c/mo>\u003cmi>R\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math> Then which of the following statements are true?\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>P : x = 0 is a point of local minima of f\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Q : x = \u003cmath>\u003cmrow>\u003cmroot>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/mroot>\u003c/mrow>\u003c/math> is a point of inflection of 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if (gtmInitScheduled) return;
gtmInitScheduled = true;

// Use scheduler API if available, fallback to requestIdleCallback
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
loadGTMScripts();
}, { priority: 'background' });
} else if ('requestIdleCallback' in window) {
requestIdleCallback(() => {
loadGTMScripts();
}, { timeout: 2000 });
} else {
// Fallback for older browsers
setTimeout(() => {
loadGTMScripts();
}, 10000);
}
}
//Important: moving this to App.js for experiment
if('yes' == 'yes'){
setTimeout(function(){lazyLoadJs(document,"script", ["https://www.googletagmanager.com/gtag/js?id=G-K11YYW39J9"],true);
window.dispatchEvent(new CustomEvent('ptupdate'));},0)
}

function loadGTMScripts() {
if (gtmInitialized) return;
gtmInitialized = true;

// Remove event listeners immediately to prevent re-triggering
document.body.removeEventListener('click', handleUserInteraction);
window.removeEventListener('scroll', handleUserInteraction, { passive: true });

// Load GTM with proper scheduling
if ('scheduler' in window && 'postTask' in window.scheduler) {
setTimeout(() => {
window.scheduler.postTask(() => {
initializeGTM();
}, { priority: 'background' });
}, 10000);
} else {
requestIdleCallback(() => {
initializeGTM();
}, { timeout: 8000 });
}
}

function initializeGTM() {
// GTM initialization code
(!function(e,t,n,o,a){
e[o]=e[o]||[];
e[o].push({"gtm.start":(new Date).getTime(),event:"gtm.js"});
var i=t.getElementsByTagName(n)[0],c=t.createElement(n),d="dataLayer"!=o?"&l="+o:"";
c.defer=!0;
c.src="https://www.googletagmanager.com/gtm.js?id="+a+d;
i.parentNode.insertBefore(c,i);
}(window,document,"script","dataLayer",false ? "GTM-WXXDPPBN" : "GTM-5FCGK6"));

// Load GPT script with delay
setTimeout(() => {
lazyLoadJs(document,"script",["https://www.googletagservices.com/tag/js/gpt.js"]);
}, 2000);
}

// Optimized user interaction handler
function handleUserInteraction() {
if (gtmInitialized) return;

// Use scheduler API for better performance
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
scheduleGTMInit();
}, { priority: 'user-visible' });
} else {
// Fallback with minimal delay
setTimeout(() => {
scheduleGTMInit();
}, 10);
}
}

// Attach optimized event listeners
document.body.addEventListener('click', handleUserInteraction, { passive: true });
window.addEventListener('scroll', handleUserInteraction, { passive: true });

window.addEventListener('message', receiveMessage, false);

function receiveMessage(event) {
if(event.data =='close_dfp'){
setTimeout(()=>{
closeBanner1();
},0)
}
}

if('0' == 1){
window.pushNotiConf = {
apiDomain : 'https://apis.shiksha.com',
deviceType : 'desktop',
fromPWA : true
};

function loadBNDependency(){if(null!=document.getElementById("pushNotificationContainer")){var e=new Date,t=e.getDate()+""+e.getMonth()+e.getFullYear()+"v10",o=Math.floor(100*Math.random()),n=function(){var e=document.createElement("script");e.src="https://js.shiksha.ws/pwa/public/unavoidable_js/shikshaPushNotification.js?"+t+o,e.async=!0,document.head.appendChild(e)},c=document.createElement("script");c.src="https://js.shiksha.ws/pwa/public/push-notification-helpers.js?"+t+o,c.async=!0,document.head.appendChild(c),"object"==typeof c.onload?c.onload=function(e){console.log("notification helper sdk loaded"),n()}:n()}}
}

window.disableInstant = true;

if (typeof history !== 'undefined' && 'scrollRestoration' in history) {
history.scrollRestoration = 'manual';
}

var spanForPDF = document.createElement('span');
spanForPDF.id = 'pageCSSPathForPDF';
spanForPDF.style = 'display:none;';
var att = document.createAttribute("csspath");
spanForPDF.setAttributeNode(att);
document.body.appendChild(spanForPDF);

function loadMultipleCSS(cssArrayList){
if(cssArrayList && cssArrayList.length>0){
cssArrayList.forEach(function(url) {
var script = document.createElement('link');
script.href = url;
script.rel = 'stylesheet';
script.type = 'text/css';
document.body.appendChild(script);
});
}
}

function loadMultipleScriptOneByOne(scriptArayList){
return new Promise( function( resolve, reject ) {
var listItem = scriptArayList.shift();
if(listItem){
var s = document.createElement('script');
var r = false;
s.type = 'text/javascript';
s.src = listItem;
s.async = true;
s.onerror = function(err) {
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
reject(err, s);
}
};
s.onload = s.onreadystatechange = function() {
// console.log(this.readyState); // uncomment this line to see which ready states are called.
if (!r && (!this.readyState || this.readyState == 'complete')) {
r = true;
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
resolve();
}
}
};
var t = document.getElementsByTagName('script')[0];
t.parentElement.insertBefore(s, t);
}else{
resolve();
}
});
}
function openChatBotRegistration(paramsObject,callBackFunction,onClose){
//versioning the files
var eVr=new Date();
var tVr= eVr.getDate()+""+eVr.getMonth()+eVr.getFullYear()+"100";
var cssArray = ['https://js.shiksha.ws/public/css/convMMP.css?v='+tVr];
var jsArray = ['https://js.shiksha.ws/public/js/jquery-3.5.1.min.js?v='+tVr,'https://js.shiksha.ws/public/js/vue.js?v='+tVr,'https://js.shiksha.ws/public/js/chatui.js?v='+tVr,'https://js.shiksha.ws/public/js/convMMP.js?v='+tVr];
// && paramsObject.entityId > 0
if(typeof(paramsObject) == 'object' && paramsObject !== null && (paramsObject.entityId > 0 || paramsObject.allowNullContId == true )){
if(typeof(loadConversationalMMP) === 'function'){
loadConversationalMMP(paramsObject,callBackFunction,onClose);
}else{
loadMultipleCSS(cssArray);
loadMultipleScriptOneByOne(jsArray).then(function (res) {
loadConversationalMMP(paramsObject,callBackFunction,onClose);
})
}
}

}
// window.blockDfpConfig = {'examPageMain': [21, 23], 'articlePage': []};
window.blockDfpConfig = {'examPageMain': [], 'articlePage': []};
window.shouldShowPEQ = true;
window.shouldShowDRC = true;
window.shouldEnableEventTracking = true;
window.shouldEnableExperimentalDFPEventTracking = false;<iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5FCGK6"
height="0" width="0" style="display:none;visibility:hidden"></iframe>!function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function(){n.callMethod?
n.callMethod.apply(n,arguments):n.queue.push(arguments)};if(!f._fbq)f._fbq=n;
n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0;
t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.appendChild(t,s)}(window,
document,'script','https://connect.facebook.net/en_US/fbevents.js');
fbq('init', '639671932819149');
fbq('track', "PageView");<img height="1" width="1" style="display:none"
src="https://www.facebook.com/tr?id=639671932819149&ev=PageView&noscript=1"
/>[118,51876,81059,26191,39658,89734,90205,55768,93281,75997,34246,17652,78029,93095,55232,82321,99774,54271,8424,90239,32215,75996,13891,97401,78305,52945,27307,25721,6429,16751,34680,57318,78002,15303,4679,15724,64827,32226,49277,68707,42397,874,93650]{"namedChunks":["desktop_header","ExamPageMain","ExamTopCard","PrepPageCTA","ExamApplyNow","ExamMenuCritical","ExamSectionalHomePage","RenderSectionHTML","SectionalECPToc","WikiLabelSectionalECP","MCQFaqTupleWrapper","AnATaggedFAQ","ANA","Icons","ExamSimilarWithTracking","AnACommon","ListWidget","MSPNewsComponent","desktop_footer","ShikshaCommonFooter"]}function stopHashScroll(event) {
event.preventDefault();
window.scrollTo(0, 0); // Scroll back to the top
}
// Function to re-enable scrolling
function enableHashScroll() {
window.removeEventListener("scroll", stopHashScroll);
}
const hash = window.location.hash;
if(hash) {
// Prevent the default scrolling behavior when the page has an anchor in the URL
window.addEventListener("scroll", stopHashScroll);
// Enable scroll when the page loads
window.addEventListener("load", enableHashScroll);
}window.dataLayer = window.dataLayer || [];
window.gtag = function (){dataLayer.push(arguments);}
window.gtag('js', new Date());
window.gtag('config', "G-K11YYW39J9");$