Question

Let $f\left(x\right)=\{\begin{array}{cc}-2,& -3\le x\le 0\\ x-2,& x<x\le 3\end{array}$ and $g\left(x\right)=f\left(|x|\right)+|f\left(x\right)|$

Which of the following statements is/are correct?
1. $g\left(x\right)$ is differentiable at $x=0$.
2. $g\left(x\right)$ is differentiable at $x=2$.
Select the correct answer using the code given below

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (A)

1 only

Given : $f\left(x\right)=\{\begin{array}{c}-2,-3\le x\le 0\\ x-2,0<x\le 3\end{array}$

$g\left(x\right)=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$

By the given $f\left(x\right)$,the graph of $f\left|x\right|and\left|f\left(x\right)\right|$ are shown in the graph.

If we add both $f\left(\left|x\right|\right)and\left|f\left(x\right)\right|$

$g\left(x\right)=f\left|x\right|+\left|f\left(x\right)\right|$

Since at $x=2$ ,the graph is sharp point.Hence it is not differentiable at $x=2$

Therefore at $x=0$ according to the graph, it is differentiable.

Therefore option A is the correct answer.