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Explain Rolle’s theorem to me

If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then there is a c in (a, b) with f′(c) = 0: the tangent is horizontal there.

Example: f(x) = x² − 4x satisfies f(1) = f(3) = −3, and indeed f′(x) = 2x − 4 vanishes at c = 2.

QuizSave

What does Rolle’s theorem guarantee?

✓A point where the derivative is zero

✕That f is constant on [a, b]

Correct. Rolle gives you that c exists, not its value.

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