Mean value theorem

Theorem 6.1 (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 ξ ∈ (a, b) such that

Proof. The geometrical meaning of the theorem is illustrated by the following figure:

The secant to a and b, has equation:

Consider the auxiliary map

From its definition: w(a) = w(b) = 0, w is continuous in [a,b] and differentiable in (a,b) so for Rolle's theorem there exists at least a point ξ ∈ (a, b) such that f'(ξ) = 0.□