Question

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

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

Answer 1

Answer: (B)

$|f{|}^2$ is differatiable everywhere

A. Even if f(x) is differentiable everywhere, |f(x)| need not be differentiable everywhere. An example being where the function changes its sign from negative to positive. $f\left(x\right)=x$ at $x=0$

B.

$|f|=$ $$\{\begin{array}{c}fiff>0\\ -fiff<0\end{array}$$

So,

$|f{|}^2=$ $$\{\begin{array}{c}{f}^{2}iff>0\\ -f^{2}iff<0\end{array}$$

Differentiating $|f{|}^2,$

$$\{\begin{array}{c}2f{f}^{\prime }iff>0\\ -2f{f}^{\prime }iff<0\end{array}$$

At $f=0,LHD=RHD=|f\left(0\right){|}^2=0,$ so function is differentiable.