Question

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

What is the value of the differential coefficient of $g\left(x\right)$ at $x=-2$?

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

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

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

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 a sharp point.Hence it is not differentiable at $x=2$

At $x=-2$ according to the graph,

The slope is in negative coordinate differntial coefficient $\Rightarrow -1$

$\Rightarrow$ The correct answer is Option A.