Legrange's Mean Value Theorem

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the right-hand side above is zero.
The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : [a, b] → R is continuous on [a, b], and that for every x in (a, b) the limit

exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the real-valued cube root function mapping x to x^1/3, whose derivative tends to infinity at the origin.
Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f(x) = e^ix for all real x. Then

f(2π) − f(0) = 0 = 0(2π − 0)

while |f′(x)| = 1.