Solution
= Solution
Suppose $f(x)=0$ and $f(x+h)/\sqrt{|h|}\to0$. Then
$$
\frac{g(x+h)-g(x)}h
=\operatorname{sgn}(h)\left(\frac{f(x+h)}{\sqrt{|h|}}\right)^2\longrightarrow0.
$$
Therefore
$$
\boxed{g'(x)=0.}
$$
This proves sufficiency of the second alternative even if $f$ itself is not <differentiable> at $x$. For example, $f(t)=|t|^{3/4}$ is <continuous> and not <differentiable> at zero, while $g(t)=|t|^{3/2}$ is <differentiable> there with derivative zero.