Rolle's Theorem

Continuous, differentiable, and f(a) = f(b) force some c ∈ (a, b) with f'(c) = 0.

The idea

Theorem (Rolle's theorem).

Let $f$ be continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and suppose $f(a) = f(b)$. Then there is some $c$ in $(a, b)$ with $f'(c) = 0$.

A smooth curve that starts and finishes at the same height must level off somewhere in between: if it ever rises it must come back down, and it cannot turn around without a point at which the tangent is horizontal.

Both hypotheses are necessary. Continuity is required on the closed interval, endpoints included, so the curve cannot jump as it reaches an end. Differentiability is required only on the open interval, because the conclusion concerns an interior point. Weaken either and the theorem can fail: a curve with a sharp corner can turn around at the corner, where no tangent exists.

The theorem asserts existence only. There may be several such $c$, and the theorem locates none of them; it does not say whether $c$ is a maximum, a minimum, or neither, only that the derivative vanishes there. That existence alone is enough to prove a great deal.

Ways to work on it

• Walkthrough. Verify Rolle's theorem on a quadratic with equal endpoint values by finding the point it promises.
• Proof. Prove Rolle's theorem from the Extreme Value Theorem and the critical-point fact.
• Practice. Find the point Rolle's theorem guarantees for a quadratic with equal endpoint values.
• Hardest. Identify when Rolle's theorem applies (and when each hypothesis fails).

Not sure where to start? Take the ten-question placement test.