Question

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

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

Answer 1

Answer: (B)

${\left|f\left(x\right)\right|}^2$ is differentiable everywhere

If $f\left(x\right)$ is differentiable, ${\left[f\left(x\right)\right]}^2={\left|f\left(x\right)\right|}^2$ is alos differentiable.

Taking $f\left(x\right)=x$, we have $f\left(x\right)=\left|f\left(x\right)\right|=\left|x\right|,$ which is not differentiable at $x=0$

Thus, $f$ is differentiable in $R\nRightarrow |f|$ is differentiable in $R$.

Also $\left(f\left|f\right|\right)\left(x\right)=f\left(x\right)\left|f\left(x\right)\right|=\{\begin{array}{c}\begin{array}{cc}-{\left[f\left(x\right)\right]}^2,& f\left(x\right)<0\end{array}\\ \begin{array}{cc}{\left[f\left(x\right)\right]}^2,& f\left(x\right)\ge 0\end{array}\end{array}$

which is differentiable in $R$ if $f$ is differentiable in $R$.