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{cases}\frac{\tan x}{x}, & x \neq 0 \\0, & x=0\end{cases}$
• C. $f ( x )=\{ x \}+\{- x \}$
• D. $f(x)=|x|-|\sin x|$

Answer 1

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

(A) $f(x)=\begin{cases}1, & x=1 \\ 0, & 0 \leq x<1 \\ x & -1 \leq x<0\end{cases} \Rightarrow $ not differentiable at $x=0$ in $(-1,1)$
(B) $ f (0)=0$ and $\underset{ x \rightarrow 0} {\text{Lim}} \frac{\tan x }{ x }=1 \Rightarrow$ not continuous at $x =0$
(C) $f ( x )=\begin{cases}1, & x \notin I \\ 0, & x \in I \end{cases} \Rightarrow $ not continuous at $x =0$
(D) $ f ( x )=\begin{cases}x -\sin x & 0 \leq x \leq 1 \\ - x +\sin x & -1 \leq x <0\end{cases} \Rightarrow$ continuous \& differentiable at $\left.x =0\right]$