Consider the function $f\left(x\right)=\left(x^3-x\right)\left|x^2-6x+5\right|, \forall x\in R$, then $f\left(x\right)$ is

Important Questions on Continuity and Differentiability

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f\left(x\right)=15–\left|x –10\right|; x\in R.$ Then the set of all values of $x$, at which the function $g\left(x\right)=f\left(f\left(x\right)\right)$ is not differentiable, is:

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

For $x\in R,f\left(x\right)=\left|\log 2-\sin x\right|$ and $g\left(x\right)=f\left(f\left(x\right)\right)$, then

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If $2y={\left({\cot }^{-1}\left(\frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x}\right)\right)}^2 \forall x\in \left(0,\frac{\pi }{2}\right)$, then $\frac{dy}{dx}$is equal to

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

The function $f\left(x\right)=\left\{\begin{array}{ll}\frac{\pi }{4}+{\tan }^{-1}x,& \left|x\right|\le 1\\ \frac{1}{2}\left(\left|x\right|-1\right),& \left|x\right|>1\end{array}\right.$ is :

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f:\left(-1, 1\right)\to R$ be a function defined by $f\left(x\right)=max\left\{-\left|x\right|, -\sqrt{1-x^2}\right\}.$ If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If $f\left(x\right)=\left[x\right]$ , where $\left[x\right]$ is the greatest integer not greater than $x$, then $f^{\text{'}}\left(1^{+}\right)$ = ….

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If the function $f\left(x\right)= \left\{\begin{array}{cc}-x,& x<1\\ a+{\cos }^{-1}\left(x+b\right),& 1 \le x \le 2\end{array}\right.$ is differentiable at $x=1$, then $\frac{a}{b}$ is equal to 

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let a function, $f:(-1,3)\to \mathrm{R}$ be defined as $f\left(x\right)=\min \left\{x\right[x],|x\left[x\right]-2|+2\},$ where $\left[x\right]$ denotes the greatest integer $\le x.$ Then $f$ is :

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If $f\left(x\right)=\left\{\begin{array}{ccc}x{e}^{-\left(\frac{1}{\left|x\right|}+\frac{1}{x}\right)};& \mathrm{i}\mathrm{f}& x\ne 0\\ 0;& \mathrm{i}\mathrm{f}& x=0\end{array}\right.$ then which of the following is correct?

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f: R\to R$ be a function such that $\left|f\left(x\right)\right|\le x^2,$ for all $x\in R.$ Then, at $x=0,$ $f$ is

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f:R\to R$ be a function defined by $f\left(x\right)=\max \left\{x,x^2\right\}$. Let $S$ denotes the set of all points in $R$, where $f$ is not differentiable. Then

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f\left(x\right)=\left\{\begin{array}{cc}max\left(\left|x\right|, x^2\right),& \left|x\right|\le 2\\ 8-2\left|x\right|,& 2<\left|x\right|\le 4\end{array}.\right.$ Let $S$ be the set of points in the interval $\left(-4, 4\right)$ at which $f$ is not differentiable. Then $S$

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f:R\to R$ be defined as $f\left(x\right)=\left\{\begin{array}{ccc}{x}^5\sin \left(\frac{1}{x}\right)+5x^2& ,& x<0\\ 0& ,& x=0\\ x^5\cos \left(\frac{1}{x}\right)+\lambda x^2& ,& x>0\end{array}\right.$. The value of $\lambda$ for which $f^{"}\left(0\right)$ exists, is___.
 

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $f:R\to R$ be differentiable at $c\in R$ and $f\left(c\right)=0$. If $g\left(x\right)=\left|f\left(x\right)\right|$, then at $x=c, g$ is:

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

The function $f\left(x\right)=e^{\left|x\right|}$ is

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If $f\left(x\right)=\left\{\begin{array}{cl}{e}^x,& x\le 0\\ \left|1-x\right|,& x>0\end{array}\right.$, then

EASY

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

At how many points between the interval $\left(-\infty ,\infty \right)$, is the function $f\left(x\right)=\sin x$ not differentiable

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $S=\left\{t\in R:f\left(x\right)=\left|x-\pi \right| \cdot \left(e^{\left|x\right|}-1\right)\sin \left|x\right| \text{is not differentiable at} t\right\}.$ Then, the set $S$ is equal to:

EASY

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

Let $S$ be the set of all points in $\left(-\pi ,\pi \right)$ at which the function, $f\left(x\right)=\min \left\{\sin x, \cos x\right\}$ is not differentiable. Then $S$ is a subset of which of the following?

HARD

Mathematics>Calculus>Continuity and Differentiability>Differentiability of a Function

If the function $g\left(x\right)=\{\begin{array}{cc}k\sqrt{x+1}\text{ , }& 0\le x\le 3\\ mx+\mathrm{2 }\text{, }& 3<x\le 5\end{array}$   is differentiable, then the value of $k+m$ is