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

(3)

.

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.