Question

Let $f\left(x\right)=\{\begin{array}{cc}\text{max}\{|x|,x^2\},& |x|\le 2\\ 8-2|x|,& 2<|x|\le 4\end{array}$
Let $S$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $S$ :

• A. is an empty set
• B. equals $\{-2,-1,1,2\}$
• C. equals $\{-2,-1,0,1,2\}$
• D. equals $\{-2,2\}$

Answer 1

The correct option is C equals $\{-2,-1,0,1,2\}$

For $\left|x\right|\le 2$

$f\left(x\right)=\{\begin{array}{cc}{x}^2,& -2\le x\le -1\\ |x|,& -1<x<1\\ x^2,& 1\le x\le 2\end{array}$
Therefore ,
For $x\in [-4,4]$

$f\left(x\right)=\{\begin{array}{cc}8+2x,& -4\le x<-2\\ x^2,& -2\le x\le -1\\ |x|,& -1<x<1\\ x^2,& 1\le x\le 2\\ 8-2x,& 2<x\le 4\end{array}$

$f$ is not differentiable at $x\in \{-2,-1,0,1,2\}$
$\therefore S=\{-2,-1,0,1,2\}$