Rolle's theorem

Rolle’s theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a)=f(b)$, then there exists $c\in(a,b)$ such that

This is a foundational consequence of the existence of a local extremum for continuous functions on compact intervals, and it is the key ingredient in the mean value theorem .