Question

Which of the following functions fail to satisfy the condition of Rolle's theorem on the interval $[-1,1]$, where $[x]$ denotes the greatest integer less or equal to $x$ and $\{x\}$ denotes the fractional part of $x$ respectively.

• A. $f(x)=|x|[x]$
• B. $f(x)=\{\begin{array}{ll}\frac{\tan x}{x},& x\ne 0\\ 0,& x=0\end{array}$
• C. $f(x)=\{x\}+\{-x\}$
• D. $f(x)=|x|-|\sin x|$

Answer 1

Answer: (A), (B), (C)

(A) $f(x)=\{\begin{array}{ll}1,& x=1\\ 0,& 0\le x<1\\ x& -1\le x<0\end{array}\Rightarrow$ not differentiable at $x=0$ in $(-1,1)$
(B) $f(0)=0$ and $\underset{x\to 0}{\mathrm{Lim}}\frac{\tan x}{x}=1\Rightarrow$ not continuous at $x=0$
(C) $f(x)=\{\begin{array}{ll}1,& x\notin I\\ 0,& x\in I\end{array}\Rightarrow$ not continuous at $x=0$
(D) $f(x)=\{\begin{array}{ll}x-\sin x& 0\le x\le 1\\ -x+\sin x& -1\le x<0\end{array}\Rightarrow$ continuous \& differentiable at $x=0]$