Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 1: Exercise 1

If $f\left(x\right)=\left\{\begin{array}{ll}{\left(1+\left|\sin x\right|\right)}^{\frac{a}{\sin x}}& ,0<x<\frac{\mathrm{\pi }}{6}\\ b& ,x=0\\ e^{\frac{\tan 2x}{\tan 3x}}& ,\frac{-\pi }{6}<x<0\end{array}\right.$, then the value of $a$ and $b$, if $f$ is continuous at $x=0$, are respectively

Let $f\left(x\right)=\left[3+4\sin x\right]$, (where $\left[·\right]$ denotes the greatest integer function). If the sum of all the values of $x$ in $\left[\pi ,2\pi \right]$, where $f\left(x\right)$ fails to be differentiable is $\frac{\mathrm{k}\pi }{2},$ then the value of $\mathrm{k}$ is

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

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

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

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

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

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