Is the absolute value of a strictly convex function also convex?

Question

If $f(x)^2$ is strictly convex, then is $|f(x)|$ convex or not?

Answer 1

No. If $f:\mathbb R\to\mathbb R$ is $f(x)=|x|^{2/3}$ then $f^2(x)=|x|^{4/3}$ is strictly convex, but $|f(x)|=|x|^{2/3}$ is not.