Rolle's Theorem — Mathematics Knowledge System

Overview

Rolle's Theorem is the special case of the Mean Value Theorem where the function takes equal values at both endpoints. It guarantees that between two points where a function has equal height, there must be at least one point where the tangent line is horizontal — a critical point.

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Formal Definition

Definition

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then:

\exists\, c \in (a,b) \text{ such that } f'(c) = 0

Rolle's Theorem

Worked Examples

Check f(1) = 1-4+3 = 0 and f(3) = 9-12+3 = 0, so f(1)=f(3) — the hypothesis holds; f is a polynomial so continuity/differentiability are automatic.
$f(1)=0,\ f(3)=0$
Set f'(x) = 0 and solve.
$f'(x) = 2x - 4 = 0 \implies x = 2$

Answer: c = 2, which lies in (1,3) as required.

Practice Problems

Difficulty 3/10

Verify Rolle's Theorem for f(x) = sin(x) on [0, π] and find the guaranteed value(s) of c.

Summary

Rolle's Theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b).
It's the special (zero-slope) case of the Mean Value Theorem, and is used as the key lemma to prove the general MVT.
Geometrically, equal function values at the endpoints force at least one horizontal tangent somewhere in between.

References

WebsiteWikipedia — Rolle's theorem