The mean value theorem (for derivatives) relates the average behavior of a function to its interior behavior. Specifically, suppose f(x) is a function continuous on [a,b] and differentiable on (a,b). Then there exists a point c in (a,b) such that f'(c) = (f(b)-f(a)) / (b-a). This natural geometric result can be used to prove that functions with vanishing derivative are constant. It is also used in one proof of one of the fundamental theorems of calculus.