The mean value theorem (explained there) is one of the most important theorems of differential calculus. This is an extension of it, discovered by Augustin-Louis Cauchy. It compares the quotient of two functions with the quotient of their derivatives, working in a manner similar to l'Hôpital's rule. In fact, the Cauchy mean value theorem can be used to prove some of l'Hôpital's rules. It also does other cute things in real analysis like that.

The typical mean value theorem tells us that as long as a real function is differentiable on a certain open interval and continuous on the endpoints, there will be a point in the interval where the tangent is parallel to the line made by connecting the endpoints.

Here is a simple proof for the Cauchy version:

Theorem:

If two functions f and g, differentiable on (a, b) and continuous on [a, b], such that f(a) = g(a) = 0,

then there exists a point c ∈ (a, b) such that the limit of f(x)/g(x) equals the limit of f'(c)/g'(c), as x approaches a in the first limit and c approaches a in the second.

By the squeeze theorem, x → a implies that c → a, since a < c < x for all x ∈ (a, b)

Proof: We invent the magical function h(x), and define it as such:

       f(b)-f(a)
h(x) = ---------(g(x)-g(a)) - (f(x)-f(a))
       g(b)-g(a)

Where did this function come from? Well h(a) = h(b) = 0 (check it!). Also, this function is nice (that is, continuous and differentiable). Because of this, Rolle's theorem tells us that there is a point c between a and b such that h'(c) = 0. So what is h'?

        f(b)-f(a)
h'(c) = ---------g'(c) - f'(c) = 0
        g(b)-g(a)

This can be solved for f'(c)/g'(c) to reconstruct the theorem. So ends the proof.