Let f: R → R be differentiable at c ∈ R and f(c) = 0 . If g(x) =...

${g}^{'}\left({c}^{+}\right) = \lim_{x \to {c}^{+}}\dfrac{\left|f\left(x\right)\right| - f\left(c\right)}{x - c} = \lim_{x \to {c}^{+}}\left|\dfrac{f\left(x\right) - f ( c )}{x - c}\right|, asx > c$

$⇒ {g}^{'}\left({c}^{+}\right) = \left|{f}^{'}\left(c\right)\right|$

$g\left({c}^{-}\right) = \lim_{x \to {c}^{-}}\dfrac{\left|f\left(x\right)\right| - f\left(c\right)}{x - c} = - \lim_{x \to {c}^{-}}\left|\dfrac{f\left(x\right) - f ( c )}{x - c}\right|, asx < c$

$⇒ {g}^{'}\left({c}^{-}\right) = - \left|{f}^{'}\left(c\right)\right|$

For $g\left(x\right)$ to be differentiable at $x = c$, ${f}^{'}\left(c\right)$ must be $0$. Else, it is non-differentiable.