Maths Applications of Derivatives

If p(x) be a polynomial of degree three that has a local maximum value 8 at x=1 and a local minimum value 4 at x=2; then p(0) is equal to
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If a and b are unit vectors, then the greatest value of √3|a+b|+|a-b| is [numerical value].

If Σᵢ₌₁ⁿ(xᵢ-a)=n and Σᵢ₌₁ⁿ(xᵢ-a)²=na, (n,a>1) then the standard deviation of n observations x₁,x₂,.,xₙ is:

Let f(x) = 3sin?x + 10sin³x + 6sin²x - 3, x ∈ [-π/6, π/2]. Then, f is

If $\mathrm{c}$ $$ is a point at which Rolle's theorem holds for the function, ${}_{}\left(\frac{{}^{}}{}\right)$ $f\left(x\right)={\mathrm{l}\mathrm{o}\mathrm{g}}_e?\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $\left[\mathrm{3,4}\right]$ $$ , where $$ , $\alpha \in R$ then ${}^{}$ $f^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(c\right)$ is equal to:

If the lines x+y=a and x-y=b touch the curve y=x²-3x+2 at the points where the curve intersects the x-axis, then a/b is equal to

If x=1 is a critical point of the function f(x)=(3x²+ax-2-a)eˣ, then

Which of the following point lies on the tangent to the curve x⁴eʸ + 2√(y+1) = 3 at the point (1,0)?

If a rectangle is inscribed in an equilateral triangle of side length 2√2 as shown in the figure, then the square of the largest area of such a rectangle is …….