KindiakMath

The Meanest Theorem in Calculus

Any self-respecting study of differential calculus must include the mean value theorem, which has ubiquitous uses in all of calculus. We will state it here and prove it in two steps: the easier step, followed by the harder step.

Theorem (Mean Value Theorem). Let f : [a,b] \to \mathbb R be continuous, and f' be differentiable on (a, b). Then there exists c \in (a, b) such that

f(b) = f(a) + f'(c)(b-a).

Proof. We will first prove the special case when f(a) = f(b), so that f'(c) = 0. This is known as Rolle’s theorem. Suppose for simplicity that f is non-constant.

Without loss of generality, there exists c_0 \in (a, b) such that f(c_0) > f(a). Since f is continuous, by the extreme value theorem, there exists c \in (a, b) such that for any x \in [a,b],

f(x) \leq f(c).

In particular, f(c) \geq f(c_0) > f(a) so that c \neq a,b. Since f is differentiable at c, we will compute its derivative f'(c) in two steps.

For the first step, by considering x-values to the left of c, since f(x) \leq f(c) and x < c,

\displaystyle f'(c) = \lim_{x \to c^-} \frac{f(x) - f(c)}{x-c} \geq 0.

For the second step, by considering x-values to the right of c, since f(x) \leq f(c) and x > c,

\displaystyle f'(c) = \lim_{x \to c^+} \frac{f(x) - f(c)}{x-c} \leq 0.

Since 0 \leq f'(c) \leq 0, we have f'(c) = 0, as desired.

Now that we have established Rolle’s theorem, we prove the mean value theorem. Define the continuous function g : [a,b] \to \mathbb R that is differentiable on (a,b) by

\displaystyle g(x) := f(x) - f(a) - \frac{f(b)-f(a)}{b-a} \cdot (x-a).

This helps us ensure that g(a) = g(b). By Rolle’s theorem, there exists c \in (a, b) such that g'(c) = 0. Alternatively, by computing g' directly,

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

Setting x = c yields

\displaystyle 0 = f'(c) - \frac{f(b) - f(a)}{b-a}\quad \iff \quad f(b) = f(a) + f'(c)(b-a),

as required.

The mean value theorem is incredibly useful in calculus, and we will be using it repeatedly as we slowly but surely define rigorously the usual functions that we have been taking for granted.

—Joel Kindiak, 20 Oct 24, 1611H

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