If $f : [a,b] \to R$ is differentiable at c, a < c < b and $f'(c) > 0$, prove that there is some x, c < x < b, such that $f(x) > f(c)$.
I'm not totally sure where to begin with this. Being that it is under the mean value section of my book, I would assume that that is relevant to the proof. My initial thought is that since the $f'(c) > 0$ we know the function is increasing at that point so you can peak a point x, c < x < b, so f(x) > f(c). I think you would need to know that the function is increasing from c to d for this to be the case though not just at the point c. So I'm not totally sure what to do from there.
Thank you anyone for the help.