Question

If $f(x)$ is differentiable everywhere then

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

Answer 1

Answer: (B)

If $f(x)$ is differentiable, $\{f(x){\}}^2.$ i.e. $|f(x){|}^2$ is also differentiable.
Taking $f(x)=x$, we have $|f|(x)=|f(x)|=|x|$, which is not differentiable at $x=0$.
Thus f is differentiable in $R\ne |f|$ is differentiable in R.
Also $(f|f|)(x)=f(x)|f(x)|=\{\begin{array}{ll}-\{f(x){\}}^2& \text{if }f(x)<0\\ \{f(x){\}}^2& \text{if }f(x)\ge 0\end{array}$
Which is differentiable in R if f is differentiable in R