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 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 .