Let be twice differentiable. If then there exists such that
This is what I have so far:
Suppose f is twice differentiable. Well, there are three cases:
(1) such that and ; or
(2) ; or
In the first case, the conclusion follows immediately from Darboux's theorem. The second case implies that is convex. And I'm stuck here. One idea is to try to prove that such that and and use Darboux's theorem again (assuming is not constant). But I don't know how to do it. I don't know if this (attempted) solution seems too complicated. Maybe there is an easier way. I'm open to suggestions.