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 ).