Embibe Experts Solutions for Exercise 7: Assignment

Which of the following function(s) not defined at $x=0$ has $/$ have removable discontinuity at $x=0$ ?

Consider a function $f\left(x\right)=\left({\cos }^{-1}x\right)\left|\sin \lambda x\right|,\lambda \in N$, $\lambda \le 100$. If for four consecutive values of $\lambda =\{{\lambda }_1$, $\left.{\lambda }_2,{\lambda }_3,{\lambda }_4\right\},f\left(x\right)$ has equal number of points of non-differentiability then

if Rolle's theorem is applicable to  $f\left(x\right)=\left[2\sin 2x\right]$ in$[a,\mathrm{b}],a\ne \mathrm{b}$ (where $\left[·\right]$ represents greatest integer function) then the positive value of $a$ and $b$ is/are

Consider functions $f\left(x\right)=\left\{\begin{array}{ll}\left|x\right|,& -1\le x\le 1\\ \left\{x\right\}, & 1<x\le 2\end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{ll}\left|x\right|,& -2<x\le 1\\ \left\{x\right\},& 1<x\le 2\end{array}\right.$ (where $\left\{·\right\}$ represents
fractional part function). Then which of the following is/are incorrect?
 

Consider a function $g\left(x\right)=\left\{\begin{array}{cc}f\left(x\right);& x\in D_f\\ 0;& x\in R-D_f\end{array}\right.$ where $f\left(x\right)={\cos }^{-1}\left|{\sin }^{-1}\left(x^2-1\right)\right|$ and $D_f=$ domain of the function $f\left(x\right),$ then which of the following is/are incorrect?

Consider a function $f\left(x\right)=sgn\left(x\right)+sgn\left(\left[x\right]\right)+\mathrm{sgn}\left(\left\{\mathrm{x}\right\}\right)$ defined in the interval $\left(-5,5\right)$, where  $\left[·\right]$and $\left\{·\right\}$ represent the greatest integer and the fractional part functions respectively. Then

Let a function $f:\left[-2,2\right]\to B$ be defined as $f\left(x\right)=\min \left(x^3,{\sin }^{-1}\left(\frac{x^3}{8}\right),{\cos }^{-1}\left(\sin x\right)\right)$. Then

If $f\left(x\right)=x^2-2\left|x\right|$ and $g\left(x\right)=\left\{\begin{array}{c}\min .\left\{f\right(t):-2\le t\le x,-2\le x\le 0\\ \max .\left\{f\right(t):0\le t\le x,0\le x\le 3\end{array}\right\}$. Then