Mean Value Theorem (Detailed)

The Mean Value Theorem is proved via Rolle's Theorem, which itself uses the Extreme Value Theorem. This layered proof demonstrates the power of compactness and continuity. The MVT is the workhorse of differential calculus, used to prove monotonicity, Lipschitz estimates, and Taylor's theorem.

Statement

Theorem5.1Mean Value Theorem

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

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

Proof

Step 1: Reduce to Rolle's Theorem. Define the auxiliary function

$g(x) = f(x) - f(a) - \frac{f(b) - f(a)}{b - a}(x - a).$

Then $g$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . Moreover,

$g(a) = f(a) - f(a) - 0 = 0,$ $g(b) = f(b) - f(a) - (f(b) - f(a)) = 0.$

By Rolle's Theorem, there exists $c \in (a, b)$ such that $g'(c) = 0$ .

Step 2: Compute $g'(c) = 0$ .

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

Thus $g'(c) = 0$ gives

$f'(c) - \frac{f(b) - f(a)}{b - a} = 0 \implies f'(c) = \frac{f(b) - f(a)}{b - a}.$

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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 $c \in (a, b)$ with $f'(c) = 0$ .

Proof

Since $f$ is continuous on $[a, b]$ (compact), by the Extreme Value Theorem, $f$ attains its maximum $M$ and minimum $m$ on $[a, b]$ .

Case 1: If $M = m$ , then $f$ is constant, so $f'(x) = 0$ for all $x \in (a, b)$ . Any $c \in (a, b)$ works.

Case 2: If $M > m$ , then at least one of $M$ or $m$ is different from $f(a) = f(b)$ . Without loss of generality, suppose $M > f(a)$ . Then $M$ is attained at some $c \in (a, b)$ (since $f(a) = f(b) < M$ ).

At this interior maximum $c$ , we have $f(c) \geq f(x)$ for all $x$ near $c$ . Thus for $h > 0$ small,

$\frac{f(c+h) - f(c)}{h} \leq 0 \quad \text{and} \quad \frac{f(c-h) - f(c)}{-h} \geq 0.$

Taking $h \to 0^+$ , $f'(c) \leq 0$ and $f'(c) \geq 0$ . Thus $f'(c) = 0$ .

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Summary

The MVT is proved by reducing to Rolle's Theorem, which uses the Extreme Value Theorem. This chain of results — Heine-Borel → EVT → Rolle → MVT — is a beautiful demonstration of the power of compactness. See Taylor's Theorem for an extension.