Question

I am repeating a closed question so other people
can see a solution of it.
Here is the problem: Suppose that
\[
f: R \to R
\]
is differentiable at c and f(c)=0.
Show that |f(x)| is differentiable at c if and only if f'(c)=0.
I will attach the solution as pdf file for people who would like to see the proof.