A Generalization of the Mean Value Theorem
Theorem: Let be a function continuous on and differentiable on . Then there is a in such that .[more]
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 .[less]
Reference: J. Tong, "The Mean-Value Theorem Generalised to Involve Two Parameters," Math. Gazette, 88(513), 2004 pp. 538–540.