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 $f'(c)=0.$

Rolle’s theorem is the key step in proving the mean value theorem and links global behavior (endpoint values) to local behavior (vanishing derivative ).

Proof sketch (optional): By the extreme value theorem , $f$ attains a maximum and a minimum on $[a,b]$. If $f$ is constant then $f'\equiv 0$. Otherwise, at least one extremum occurs at an interior point $c\in(a,b)$, and differentiability forces $f'(c)=0$ at an interior local extremum .