Rolle's Theorem: statement, conditions and a solved example

Statement

If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then there is at least one c in (a, b) with f′(c) = 0.

Why every condition is needed

• Not continuous: f(x) = x for 0 ≤ x < 1 and f(1) = 0. Then f(0) = f(1) but f′(x) = 1 everywhere in (0, 1).
• Not differentiable: f(x) = |x| on [−1, 1]. f(−1) = f(1) = 1, but f′ is ±1 and is not defined at 0, so f′ is never 0.
• f(a) ≠ f(b): f(x) = x on [0, 1] has f′(x) = 1, never 0.

Meaning on the graph

Somewhere between a and b the graph has a horizontal tangent.

Solved example

Check Rolle's theorem for f(x) = x³ − 3x on [0, √3].

• f is a polynomial, so it is continuous and differentiable everywhere.
• f(0) = 0 and f(√3) = 3√3 − 3√3 = 0, so f(0) = f(√3).
• f′(x) = 3x² − 3 = 0 gives x = ±1. Only c = 1 lies in (0, √3).

So Rolle's theorem holds with c = 1.