Cauchy's mean value theorem

The mean value theorem for two functions at once — for a parametric curve $(g(t), f(t))$. It is the lemma from which L'Hôpital's rule follows.

Apply Rolle to $h(x) = (f(b) - f(a))\,g(x) - (g(b) - g(a))\,f(x)$, which takes the same value at $a$ and $b$.