---
tag: 0BARQY52N5
kind: statement
role: theorem
name: Mean value theorem
aliases: [lagrange-mean-value-theorem]
msc: [26A24]
uses: [0194DY35BG]
sources:
  - url: https://en.wikipedia.org/wiki/Mean_value_theorem
    locator: Statement
same_as:
  wikidata: Q189136
  wikipedia: Mean value theorem
---

## Statement

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

## Remarks

Apply [[Rolle's theorem|0194DY35BG]] to $g(x) = f(x) - \frac{f(b) - f(a)}{b - a} (x - a)$, which has $g(a) = g(b)$.
It turns bounds on $f'$ into bounds on $f$. It gives [[zero derivative implies constant|003QDZ4T16]] and the [[monotonicity tests|07J5X2T1ZX]].
Mathlib states it as `exists_deriv_eq_slope`.

## History

Augustin-Louis Cauchy stated and proved the theorem in its modern form in 1823.

## Explanations

### L1 human

Somewhere between $a$ and $b$, the tangent line is parallel to the chord from $(a, f(a))$ to $(b, f(b))$.
If a car travels 100 km in one hour, at some moment its speed is exactly 100 km per hour.