> [!theorem] > > Let $f: [a, b] \to \real$ be a [[Function|function]] that is [[Continuity|continuous]] on $[a, b]$ and [[Derivative|differentiable]] on $(a, b)$. If $f(a) = f(b)$, then $\exists c \in (a, b)$ where the [[Derivative|derivative]] $f^\prime(c) = 0$. > > *Proof*. Since $f$ is continuous over a [[Compactness|compact]] [[Set|set]] ([[Heine-Borel Theorem]]), $f$ has a [[Extrema|maximum]] and minimum ([[Extreme Value Theorem|extreme value theorem]]). If both the maximum and minimum occur at both of the endpoints, then $f$ is constant and $f^\prime(x) = 0 \forall x \in (a, b)$. Otherwise, the maximum and minimum occur in $(a, b)$, and $\exists c \in (a, b): f^\prime(c) = 0$ ([[Interior Extremum Theorem|interior extremum theorem]]).