Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 3: Exercise-3

Let $f, g, h$ be real valued functions defined on the interval $\left[0,1\right]$ by $f\left(x\right)=e^{x^2}+e^{-x^2}, g\left(x\right)=x{e}^{x^2}+e^{-x^2}$ and $h\left(x\right)=x^2{e}^{x^2}+e^{-x^2}$. If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ and $h$ on $\left[0,1\right]$, then

The slope of the tangent to the curve ${\left(y-x^5\right)}^2=x{\left(1+x^2\right)}^2$ at the point $(1,3)$ is

Let $f:R\to (0,\infty )$ and $g:R\to R$ be twice differentiable functions such that $f^{"}$ and $g^{"}$ are continuous functions on $R$. Suppose $f^{\text{'}}\left(2\right)=g\left(2\right)=0,f^{"}\left(2\right)\ne 0$ and $g^{\text{'}}\left(2\right)\ne 0$, If $\underset{x\to 2}{\lim }\frac{f\left(x\right)g\left(x\right)}{f^{\text{'}}\left(x\right)g^{\text{'}}\left(x\right)}=1$, then

If $f: R\to R$ is a differentiable function such that $f^{\text{'}}\left(x\right)>2f\left(x\right)$ for all $x\in R,$ and $f\left(0\right)=1$ then

Let $f:R\to R$ be defined by $f\left(x\right)=\left\{\begin{array}{ll}k-2x,& \text{if} x\le -1\\ 2x+3,& \text{if} x>-1\end{array}\right.$. If $f$ has a local minimum at $x=-1$, then a possible value of $k$ will be

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side$=x$ units and a circle of radius$=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then

Consider $f\left(x\right)={\tan }^{-1} \left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), x\in \left(0,\frac{\pi }{2}\right)$. A normal to $y=f\left(x\right)$ at $x=\frac{\pi }{6}$ also passes through the point 

The normal to the curve $y\left(x-2\right)\left(x-3\right)=x+6$ at the point where the curve intersects the $y-$axis passes through the point :