Different form in Rolle's theorem I am given a high school question which I find I am unable to solve. The question is as follow:
Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is differentiable everywhere. For some $a,b\in\mathbb{R}, a<b, f'(a)<0 $ and $ f'(b)>0$. Prove there is $c$ in $a,b$ such that $f'(c)=0$.
My first insight is of course, if $f'$ is continuous, then the result comes obvious as $f'$ must pass through $0$. However, immediately I realized that $f$ is just differentiable, which does not imply $f'$ is continuous. But I find it is also near to impossible to construct a counter example using example from google, like well known $x^2sin(\frac{1}{x})$...

Thanks to Qiaochu Yuan, the answer is actually very straight forward by considering extreme value theorem on $f$ rather than playing around with $f'$. Guaranteed with an extreme, its $f'=0$
