Question

If $f\left(x\right)$ is a differentiable everywhere then

• A. $|f\left(x\right)|$ is differentiable everywhere
• B. $|f{|}^2$ is differentiable everywhere
• C. $f|f|$ is not differentiable at some point
• D. None of these

Answer 1

The correct option is B $|f{|}^2$ is differentiable everywhere

$f\left(x\right)$ is a differentiable, $\left\{f\right(x){\}}^2=|f(x){|}^2$ is also differentiable.
Substitute of $f\left(x\right)=x,$ then
Therefore $|f\left(x\right)|=|x|$ is not differentiable at $x=0$
Thus $f$ is differentiable in $R\Rightarrow |f|$ is not differentiable in $R.$
$f\left(x\right)|f\left(x\right)|=\{\begin{array}{c}-\left\{f\right(x\left){\}}^2\text{if}f\right(x)<0\\ \\ \left\{f\right(x\left){\}}^2\text{if}f\right(x)>0\end{array}$

The above expression is differentiable in $R$ if $f$ is differentiable in $R.$