If $f\left(x\right)=x\left[\sqrt{x}-\sqrt{x+1}\right],$ then

Important Questions on Continuity and Differentiability

HARD

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 1>Q 36

If $f\left(x\right)=\left\{\begin{array}{l}{e}^x+ax,x<0\\ b\left(x-1\right){}^2,x\ge 0\end{array}\right.$ is differentiable at $x=0,$ then $(a,b)$ is

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 1>Q 37

Let $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$ and $f\left(x\right)=x^{2}g\left(x\right)$ for all $x,y\in R$ where  $g\left(x\right)$ is continuous function. Then $f^{\text{'}}\left(x\right)$ is equal to

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 1>Q 38

If $f$ is a real valued differentiable function satisfying $\left|f\left(x\right)-f\left(y\right)\right|\le {\left(x-y\right)}^2, x, y\in R$ and $f\left(0\right)=0$, then $f\left(1\right)$ equals

HARD

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 1>Q 39

The function $f:R/\left\{0\right\}\to \to R$ given by $f\left(x\right)=\frac{1}{x}-\frac{2}{e^{2x}-1}$ can be made continuous at $x=0$ by defining $f\left(0\right)$ as

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 1>Q 40

If $f\left(x\right)$ is differentiable and $f^{\text{'}}\left(x\right)>0$, then the value of $\underset{x\to 0}{\lim }\frac{f\left(x^2\right)-f\left(x\right)}{f\left(x\right)-f\left(0\right)}$ is

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 1

The number of points at which the function $f\left(x\right)=\frac{1}{x-\left[x\right]}$, where $\left[·\right]$ denotes the greatest integer function, is not continuous is

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 2

If $f\left(x\right)=2 x$ and  $g\left(x\right)=\frac{x^2}{2}+1,$ then which of the following can be a discontinuous function?

MEDIUM

AP EAPCET

IMPORTANT

Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 3

If $f\left(x\right)=\left\{\begin{array}{cl}\frac{1-\sqrt{2}\sin x}{\pi -4x},& \mathrm{if} x\ne \frac{\pi }{4}\\ a& , \mathrm{if} x=\frac{\pi }{4}\end{array}\right.$ is continuous at $x=\frac{\pi }{4}$, then $a$ is equal to