Solution
= Solution
\b[No.] Both $f(x)=x$ and $g(x)=-x$ are <affine functions>, hence <convex functions>. Their product $h(x)=-x^2$ is not <convex>: at the midpoint of $-1$ and $1$, $h(0)=0>\tfrac12h(-1)+\tfrac12h(1)=-1$. Thus <products of convex functions need not be convex>, even for smooth functions on all of $\mathbb R$.