# Proving theorems and special cases (Part 15): The Mean Value Theorem

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve seen that a conjecture could be true for the first 40 cases or even the first $10^{316}$ cases yet not always be true. We’ve also explored the computational evidence for various unsolved problems in mathematics, noting that even this very strong computational evidence, by itself, does not provide a proof for all possible cases.

However, there are plenty of examples in mathematics where it is possible to prove a theorem by first proving a special case of the theorem. For the remainder of this series, I’d like to list, in no particular order, some common theorems used in secondary mathematics which are typically proved by first proving a special case.

6. Theorem (Mean Value Theorem). If $f$ is a continuous function on the interval $[a,b]$ which is differentiable on the interior $(a,b)$, then there is a point $c \in (a,b)$ so that

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

In other words, there is a point $c$ in $(a,b)$ so that the slope of the tangent line at $c$ is the same as the slope of the line segment connecting the endpoints.

This is a consequence of the following lemma.

Lemma (Rolle’s Theorem). If $f$ is a continuous function on the interval $[a,b]$ which is differentiable on the interior $(a,b)$ so that $f(a) = 0$ and $f(b) = 0$, then there is a point $c \in (a,b)$ so that $f'(c) = 0$.

Notice that Rolle’s Theorem is really a special case of the Mean Value Theorem: if $f(a) = 0$ and $f(b) = 0$, then the right-hand side of the conclusion of the Mean Value Theorem becomes

$\displaystyle \frac{f(b)-f(a)}{b-a} = \displaystyle \frac{0-0}{b-a} = 0$,

thus matching the conclusion of Rolle’s Theorem.

I won’t type out the proofs of Rolle’s Theorem and the Mean Value Theorem here, since Wikipedia has already done that very well. Suffice it to say that Rolle’s Theorem logically comes first, and then the Mean Value Theorem can be proven using Rolle’s Theorem. The main idea is to assume that the function $f$ satisfies the hypotheses of the Mean Value Theorem and then define

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

It’s straightforward to show that $g$ satisfies the hypotheses of Rolle’s Theorem and conclude that there must be a point so that $g'(c) = 0$, from which we obtain the conclusion of the Mean Value Theorem.