A Generalization of the Mean Value Theorem

Theorem: Let be a function continuous on and differentiable on . Then there is a in such that .

Proof: the theorem follows by applying Rolle's theorem to the auxiliary function

Here is a geometric interpretation: The triangle formed by the axis, the tangent line through , and the secant line through and the point is an isosceles triangle (the green triangle). Therefore the slopes of the two sides not on the axis are and .

The example used is the function .