Proof of Rolle's theorem

The proof of Rolle’s theorem rests on Weierstrass’s theorem, which guarantees the existence of an absolute maximum and minimum for a continuous function on a closed and bounded interval.

Proof

By Weierstrass, $f$ continuous on $[a,b]$ admits a maximum $M$ and a minimum $m$. If $M=m$, $f$ is constant and $f'(c)=0$ for every $c$. Otherwise, at least one of $M$ and $m$ is attained at an interior point $c\in(a,b)$: indeed, since $f(a)=f(b)$, if the maximum were attained only at the endpoints we would have $M=f(a)=f(b)=m$, a contradiction. At $c$ the derivative vanishes, because the vanishing of the derivative is a necessary condition for an interior extremum. $\blacksquare$

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Topics: Calculus theorems
Concepts: Rolle’s theorem · Weierstrass’s theorem
Skills: Proving
People: Michel Rolle · Karl Weierstrass