Question

Let $f\left(x\right)=\{\begin{array}{c}-1,-2\le x<0\\ x^2-1,0\le x\le 2\end{array}$ and $g\left(x\right)=|f\left(x\right)|+f\left(|x|\right)$. Then , in the interval $(-2,2),g$ is:-

• A. differentiable at all points
• B. not differentiable at two points
• C. not continuous
• D. not differentiable at one point

Answer 1

The correct option is D not differentiable at one point

$|f\left(x\right)|=\{\begin{array}{cc}1,& -2\le x<0\\ 1-x^2,& 0\le x<1\\ x^2-1,& 1\le c\le 2\end{array}$
and $f\left(|x|\right)=x^2-1,x\in [-2,2]$
Hence $g\left(x\right)=\{\begin{array}{cc}{x}^2,& x\in [-2,0)\\ 0,& x\in [0,1)\\ 2(x^2-1),& 1\le c\le 2\end{array}$
It is not differentiable at $x=1$