...solved!
If $\displaystyle f(x) = |x|$, then $\displaystyle f'(x) = \frac{|x|}{x}$.
Therefore, if $\displaystyle f(x) = |x|^\frac{3}{2}$, then $\displaystyle f'(x) = \frac{3}{2}|x|^\frac{1}{2} \times \frac{|x|}{x}$.
$\displaystyle f'(x)$ fails to exist at $\displaystyle x = 0$ (because of division by zero). It exists at all other values of $\displaystyle x$.
Likewise, $\displaystyle f''(x)$ is undefined at $\displaystyle x = 0$ but defined for all other values of $\displaystyle x$.