Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 4 / b / Solution

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 $1$, $h(0)=0>\frac{1}{2}h(-1)+\frac{1}{2}h(1)=-1$. Thus products of convex functions need not be convex, even for smooth functions on all of $\mathbb{R}$.