Mean Value Theorem

The Mean Value Theorem (MVT) states that for a differentiable function on an interval, there exists a point where the instantaneous rate of change (derivative) equals the average rate of change. This result connects local and global properties of functions and is the foundation for Taylor's theorem, L'Hôpital's rule, and inequalities involving derivatives.

Statement

Theorem5.1Mean Value Theorem

Let $f : [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$ . Then there exists $c \in (a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}.$

RemarkGeometric interpretation

The MVT says the secant line connecting $(a, f(a))$ and $(b, f(b))$ is parallel to the tangent line at some point $c \in (a, b)$ . There is a point where the instantaneous slope equals the average slope.

ExampleLinear functions

For $f(x) = mx + b$ , we have $f'(x) = m$ everywhere. The MVT is satisfied by any $c \in (a, b)$ : $f'(c) = m = (f(b) - f(a))/(b - a)$ .

Rolle's Theorem

Theorem5.2Rolle's Theorem

If $f : [a, b] \to \mathbb{R}$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f(a) = f(b)$ , then there exists $c \in (a, b)$ such that $f'(c) = 0$ .

RemarkRolle's theorem is a special case

Rolle's theorem is the MVT with $f(a) = f(b)$ (so the average rate of change is zero). Conversely, the MVT follows from Rolle's theorem applied to $g(x) = f(x) - Lx$ where $L = (f(b) - f(a))/(b - a)$ .

ExampleRolle's theorem for f(x) = x² - 1

Let $f(x) = x^2 - 1$ on $[-1, 1]$ . Then $f(-1) = f(1) = 0$ , so by Rolle's theorem, there exists $c \in (-1, 1)$ with $f'(c) = 0$ . Indeed, $f'(x) = 2x$ , so $c = 0$ .

Applications

ExampleCharacterizing increasing functions

$f$ is increasing on $[a, b]$ if and only if $f'(x) \geq 0$ for all $x \in (a, b)$ .

Proof: If $f'(x) \geq 0$ , then for $x < y$ , by MVT, there exists $c \in (x, y)$ with $f(y) - f(x) = f'(c)(y - x) \geq 0$ . Thus $f(x) \leq f(y)$ .

ExampleFunctions with zero derivative are constant

If $f'(x) = 0$ for all $x \in (a, b)$ , then $f$ is constant on $[a, b]$ .

Proof: For $x < y$ in $[a, b]$ , by MVT, $f(y) - f(x) = f'(c)(y - x) = 0$ , so $f(y) = f(x)$ .

ExampleLipschitz inequality

If $|f'(x)| \leq M$ for all $x \in (a, b)$ , then $|f(x) - f(y)| \leq M|x - y|$ for all $x, y \in [a, b]$ (Lipschitz continuity).

Proof: By MVT, $|f(x) - f(y)| = |f'(c)||x - y| \leq M|x - y|$ .

Summary

The Mean Value Theorem is central in differential calculus:

Connects average rate of change to instantaneous rate of change.
Rolle's theorem is a special case (when $f(a) = f(b)$ ).
Applications: characterizing monotonicity, Lipschitz continuity, proving inequalities.

See Mean Value Theorem proof and Taylor's Theorem for extensions.