The **mean value theorem** states that in a closed interval, a function has at least one point where the slope of a tangent line at that point (i.e. the derivative) is equal to the average slope of the function (or the secant line between the two endpoints).

Ergo:

on a closed interval has a derivative at point , which has an equivalent slope to the one connecting and .

Therefore, the derivative equals the slope formula:

There are three formulations of the mean value theorem:

## Rolle's Theorem

Rolle's theorem states that for a function that is continuous on and differentiable on :

If then

### Proof

By the Weierstrass Theorem, the function has two extrema in , say a minimum and a maximum . There are two cases:

(i) If then (by the condition for Rolle's Theorem to hold). However, (as it's a minimum) and (as it's a maximum). So .

Therefore is constant on , so its derivative is 0 everywhere, so there certainly exists a with .

(ii) If the above case does not happen, then . So take , and as it is an extremum .

## Lagrange's Mean Value Theorem

Lagrange's mean value theorem, sometimes just called the mean value theorem, states that for a function that is continuous on and differentiable on :

### Proof

Rather than prove this theorem explicitly, it is possible to show that it follows directly from Rolle's theorem. As we have already proved Rolle's, this is enough.

Define a function

Observe

And

Note that by the algebra of continuous and differentiable functions, satisfies the conditions for Rolle's Theorem.

So by the theorem, ,

So ,

i.e. .

Note also that Rolle's Theorem is a special case of Lagrange's MVT, where .

## Cauchy's Mean Value Theorem

Cauchy's mean value theorem states that for two functions that are continuous on and differentiable on :

### Proof

Note: because saying the opposite will apply Rolle's Theorem that .

Again we can show this follows from Rolle's Theorem:

Define a function

Observe

And

Again, by the algebra of continuous and differentiable functions, also satisfies the other conditions for Rolle's Theorem.

So by the theorem,

So

i.e. .

Note that Lagrange's MVT (and therefore also Rolle's Theorem) is just a special case of Cauchy's MVT, where you take .