Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 3: Exercise-3

Let $f_1:R\to R, f_2:\left[0,\mathrm{\infty }\right)\to R, f_3:R\to R$ and $f_4:R\to \left[0,\mathrm{\infty }\right)$ be defined by
$f_1\left(x\right)=\left\{\begin{array}{lll}\left|x\right|& \text{ if }& x<0\\ e^x& \text{ if }& x\ge 0\end{array}\right.$
$f_2\left(x\right)=x^2$
$f_3\left(x\right)=\left\{\begin{array}{ccc}\sin x& \text{ if }& x<0\\ x& \text{ if }& x\ge 0\end{array}\right.$
and $f_4\left(x\right)=\left\{\begin{array}{ll}{f}_2\left(f_1\left(x\right)\right)& \text{ if } x<0\\ f_2\left(f_1\left(x\right)\right)-1& \text{ if } x\ge 0\end{array}\right.$

Let $g:R\to R$ be a differentiable function with $g\left(0\right)=0, g^{\text{'}}\left(0\right)=0$ and $g^{\text{'}}\left(1\right)\ne 0$. Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\left|x\right|}g\left(x\right),& x\ne 0\\ 0,& x=0\end{array}\right.$ and $h\left(x\right)=e^{\left|x\right|}$ for all $x\in R.$ Let $\left(foh\right)\left(x\right)$ denotes $f\left\{h\left(x\right)\right\}$ and $\left(hof\right)\left(x\right)$ denotes $h\left\{f\left(x\right)\right\}$. Then, which of the following is/are true?

Let $f:\left[-\frac{1}{2},2\right]\to \mathrm{R}$ and $g:\left[-\frac{1}{2},2\right]\to R$ be functions defined by $f\left(x\right)=\left[x^2-3\right]$ and $g\left(x\right)=\left|x\right|f\left(x\right)+\left|4x-7\right|f\left(x\right)$, where $\left[y\right]$ denotes the greatest integer less than or equal to $y$ for $y\in R.$ Then

Let $\left[x\right]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\mathrm{\pi }\right(x+\left[x\right]\left)\right)$ is discontinuous?

Let $f_1:R\to R,f_2:,\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R,f_3:\left(-1,e^{\frac{\pi }{2}}-2\right)\to R$ and $f_4:R\to R$ be functions defined by

(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$

(ii) $f_2\left(x\right)=\left\{\begin{array}{ccc}\frac{\left|\sin x\right|}{{\tan }^{-1}x}& \text{ if }& x\ne 0\\ 1& \text{ if }& x=0\end{array}\right.$ where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

(iii) $f_3\left(x\right)=\left[\sin \left({\log }_e(x+2)\right)\right],$ where for $t\in R,\left[t\right]$ denotes the greatest integer less than or equal to$t$

(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right)& \text{ if }x\ne 0\\ 0& \text{ if }x=0\end{array}\right.$

Define $F\left(x\right)$ as the product of two real functions $f_1\left(x\right)=x, x\in R,$ and $f_2\left(x\right)=\left\{\begin{array}{ccc}\sin \frac{1}{x},& \text{ If }& \mathrm{x}\ne 0\\ 0,& \text{ If }& x=0\end{array}\right.$ as follows

$F\left(x\right)=\left\{\begin{array}{ll}{f}_1\left(x\right)·f_2\left(x\right)& \text{ If } x\ne 0\\ 0,& \begin{array}{cc}\text{ If }& x=0\end{array}\end{array}\right.$

Statement -$1: F\left(x\right)$ is continuous on $R.$

Statement - $2: f_1\left(x\right)$ and $f_2\left(x\right)$ are continuous on $R.$

Consider the function, $f\left(x\right)=|x-2|+|x-5|, x\in R.$

Statement- $1:\hspace{0.17em}{f}^{\text{'}}\left(4\right)=0$

Statement- $2: f$ is continuous in $[2, 5],$ differentiable in $(2, 5)$ and $f\left(2\right)=f\left(5\right).$

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