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.
By sticking to your reasoning:
(1) Case solved
(2) Assume that
use the fact that a convex function always lies above its tangent(s) and show that there is at least one tangent with a strictly positive slope. Hence will tend to on the right which is in contradiction with the initial assumption on . Hence
(3) Assume that
use the fact that a concave function always lies below its tangent(s) and show that there is at least one tangent with a strictly negative slope. Hence will tend to on the right which is in contradiction with the initial assumption on . Hence
For (2)-(3) (existence of such tangent):
Since , . So if we take , the strict positivity (resp. negativity) of means is strictly increasing (resp. strictly decreasing). Hence (resp. ). We've found a tangent with a strictly positive (resp. strictly negative) slope.