Explanation

(a)
Given, $f(x) = |x|^{3} = \left\{ \begin{matrix}
0, & x = 0 \\
x^{3}, & x > 0 \\
- x^{3}, & x < 0
\end{matrix} \right.\$
Now, $Rf^{'}(0) = \lim_{h \rightarrow 0}\frac{f(h) - f(0)}{h}$
$= \lim_{h \rightarrow 0}\frac{h^{3} - 0}{h} = 0$
and $Lf^{'}(0) = \lim_{h \rightarrow 0}\frac{f( - h) - f(0)}{- h}$
$= \lim_{h \rightarrow 0}{\frac{- h^{3} - 0}{- h} = 0}$
$\because Rf^{'}(0) = Lf^{'}(0) = 0$
$\therefore f^{'}(0) = 0$