Question

Let $f:R\to R$ be a function defined by $f\left(x\right)=max\{x,x^3\}.$ Then the set of all points where $f$ is not differentiable, is

• A. $\{-1,0\}$
• B. $\{-1,0,1\}$
• C. $\{0,1\}$
• D. $\{-1,1\}$

Answer 1

The correct option is B $\{-1,0,1\}$

$f\left(x\right)=\{\begin{array}{cc}x,& -\infty <x\le -1\text{and}0\le x\le 1\\ x^3,& -1<x<0\text{and}x>1\end{array}$

At $x=-1,$
$LHD=1,RHD=3x^2=3$

At $x=0,$
$LHD=3x^2=0,RHD=1$

At $x=1,$
$LHD=1,RHD=3x^2=3$
Hence $f$ is not differentiable at $-1,0,1$

Alternative solution :

Clearly, we can observe from the graph that the function $f$ is not differentiable at $x=-1,0,1$