---
tag: 0194DY35BG
kind: statement
role: theorem
name: Rolle's theorem
msc: [26A24]
uses: [0WS1A8E2P9, 0PCVBRV9VJ]
sources:
  - url: https://en.wikipedia.org/wiki/Rolle%27s_theorem
    locator: Statement
same_as:
  wikidata: Q193286
  wikipedia: Rolle's theorem
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$.
Then there is $c \in (a, b)$ with $f'(c) = 0$.

## Remarks

By the [[extreme value theorem|0WS1A8E2P9]], $f$ has a maximum and a minimum on $[a, b]$. If both occur at the endpoints, $f$ is constant. Otherwise one occurs at an interior point $c$, and [[Fermat's theorem|0PCVBRV9VJ]] gives $f'(c) = 0$.
Differentiability inside the interval is needed. The function $|x|$ on $[-1, 1]$ has no point with derivative $0$.
Mathlib states it as `exists_deriv_eq_zero`.

## History

Michel Rolle proved a version for polynomials in 1691. Cauchy gave a proof of the general theorem in 1823.

## Explanations

### L1 human

A smooth graph that starts and ends at the same height must turn around somewhere.
At the turning point, the tangent is horizontal.