Corollary 1 — Rolle’s Theorem
If [latex]f[/latex] is continuous on the interval [latex][a,b][/latex] and differentiable on the interior [latex](a,b)[/latex], and if [latex]f(a)=f(b)[/latex], then there exists a point [latex]c[/latex] in [latex](a,b)[/latex] with [latex]f'(c)=0[/latex].
Proof
If there were no such point [latex]c[/latex], then the interval [latex][a,b][/latex] would satisfy the conditions of the preceding theorem. The function [latex]f[/latex] would be strictly monotone over [latex][a,b][/latex] and the values [latex]f(a)[/latex] and [latex]f(b)[/latex] would not be equal. This contradicts the assumption [latex]f(a)=f(b)[/latex].
Remark
One may also note that Rolle’s theorem follows from the first paragraph of the proof of the theorem.
