Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 1: Assam CEE 2017

The normal to the curve $x=a(\cos \theta +\theta \sin \theta ), y=a(\sin \theta -\theta \cos \theta )$ at any point $\theta$ is such that

$y=\left\{x\right(x-3){\}}^2$ increases for all values of $x$ lying in the interval

The normal to the curve $2x^2+y^2=12$ at the point $(2,2)$ cuts the curve again at:

If $f$ be a continuous function on $[0,1]$, differentiable in $(0,1)$ such that $f\left(1\right)=0$, then there exists some $c\in (0,1)$ such that:

Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\text{'}\text{'}}\left(x\right)<0\forall x\in (0,2).$ Then $g\left(x\right)$ increases in:

The lengths of the sides of a rectangle of greatest area that can be inscribed in the ellipse $9x^2+16y^2=144$ are

Let $f$ and $g$ be increasing and decreasing functions respectively from $[0,\infty )$ to $[0,\infty )$ and $h\left(x\right)=f\left\{g\left(x\right)\right\}$. If $h\left(0\right)=0$, then $h\left(x\right)-h\left(1\right)$ is

The tangent at $\left(1,7\right)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0$  at