The composition of Convex functions?

Question

My professor gave us the following question:

Refute (with a simple example): Let ,:ℝ→ℝ be two convex functions. The composition ℎ≜∘ (that is, ℎ()=(())) is also a convex function.

But from what I can read online the composition of two convex functions is convex as well, what am I missing here?

The composition of two convex functions is convex

@Aig ohh the title is misleading thanks, so in my counter example I should look for deceasing g but wasn't successful finding one, can u guide me? (I will do the proof) — David, Mar 07 '24 at 04:31

score 2 · Answer 1 · answered Mar 07 '24 at 06:15

2

Let $f$ and $g$ be $f(x)=-x$, $g(x)=x^2$. Then $f$ and $g$ are convex (since they are twice continuously differentiable and its second derivatives are $\geq 0$). However, $f(g(x))=-x^2$ is not convex.

answered Mar 07 '24 at 06:15

Iván G M

471

The composition of Convex functions?

1 Answers1