Applications of Derivatives

f (x) = (3x - 7)x²/³
⇒ f (x) = 3x? /³ - 7x²/³
⇒ f' (x) = 5x²/³ - 14/ (3x¹/³)
= (15x - 14) / (3x¹/³) > 0

∴ f' (x) > 0 ∀x ∈ (-∞, 0) U (14/15, ∞)

f(3)=f(4) ⇒ α=12
f'(x) = (x²-12)/(x(x²+12))
∴ f'(c)=0 ⇒ c=√12
∴ f''(c) = 1/12

0.50
 y = x²-3x+2, x+y=a, x-y=b
x?=2 x?=1
y? = 4-6+2 = 0 y?=0
(2,0)
(1,0)
b=2
a=1
∴ a/b = 1/2 = 0.5

e? y'x? + 4x³e? + 2y' / (2√ (y+1) = 0 at (1,0)
y' + 4 + y' = 0 ⇒ y' = -2
equation of tangent at (1,0) is 2x + y - 2 = 0
So option (C) is correct.

The Class 12 Applications of Derivatives chapter is mainly focused on the usage of derivatives rather a lot formulas. Students can check the important formulae below;

• Rate of Change of a Quantity

If $y = f (x)$y=f (x), then the rate of change of $y$y with respect to $x$x is:

$\frac {dy} {dx}$

• Increasing and Decreasing Functions

• A function $f (x)$ is increasing on interval $I$ if $f' (x) > 0$for all $x \in I$  and decreasing if $f' (x) < 0$ for all $x \in I$