Lagrange's Mean Value Theorem

Drive 120 km in two hours. Your average speed was 60 km/h. At some instant during the trip, the speedometer must have read exactly 60 km/h. Lagrange’s Mean Value Theorem is this fact stated for any differentiable function.

Statement

Theorem (Mean Value Theorem / Lagrange’s Finite Increment Theorem). Let $f : [a, b] \to \mathbb{R}$. If

1. $f$ is continuous on $[a, b]$, and
2. $f$ is differentiable on $(a, b)$,

then there exists $c \in (a, b)$ such that

$$f(b) - f(a) \;=\; f'(c)\,(b - a). \tag{1}$$

Proof

The idea is to subtract from $f$ the linear function whose graph is the secant line through $(a, f(a))$ and $(b, f(b))$, turning the problem into one where Rolle’s theorem applies.

Define the auxiliary function

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

Then:

• $g$ is continuous on $[a, b]$ and differentiable on $(a, b)$ (inherited from $f$).
• $g(a) = 0$ and $g(b) = f(b) - f(a) - (f(b)-f(a)) = 0$.

By Rolle’s Theorem, there exists $c \in (a, b)$ with $g'(c) = 0$. Computing:

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

so $g'(c) = 0$ gives $f'(c) = \dfrac{f(b)-f(a)}{b-a}$, which is exactly $(1)$. $\square$

Geometric meaning

The right-hand side of $(1)$ is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. The theorem says the tangent line at some interior point is parallel to that secant. In other words, the instantaneous rate of change equals the average rate of change at some point.

Corollaries

Constant functions

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

Proof. For any $x \in [a, b]$, apply the MVT to $[a, x]$: $f(x) - f(a) = f'(c)(x - a) = 0$, so $f(x) = f(a)$. $\square$

Monotone functions

Corollary. If $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is strictly increasing on $[a, b]$.

Proof. For $a \leq x_1 < x_2 \leq b$, apply the MVT to $[x_1, x_2]$: $f(x_2) - f(x_1) = f'(c)(x_2 - x_1) > 0$. $\square$

The analogue with $f' < 0$ gives strict decrease; $f' \geq 0$ (or $\leq 0$) gives non-strict monotonicity.

Lipschitz bound

Corollary. If $|f'(x)| \leq M$ for all $x \in (a, b)$, then $|f(x_2) - f(x_1)| \leq M|x_2 - x_1|$ for all $x_1, x_2 \in [a, b]$.

This is the Lipschitz condition with constant $M$, widely used to control approximation errors.

Summary

• Mean Value Theorem: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f(b) - f(a) = f'(c)(b-a)$ for some $c \in (a,b)$.
• Proof: apply Rolle’s theorem to $g(x) = f(x) - \text{(secant line)}$.
• Corollaries: zero derivative → constant; positive derivative → strictly increasing; bounded derivative → Lipschitz.
• The theorem links global change $f(b)-f(a)$ to a local quantity $f'(c)$, making it the workhorse for proving properties of differentiable functions.