Here we illustrate the [b]Mean Value Theorem[/b]: if a function [math]f[/math] satisfies the two conditions[br][list][*][color=#980000][math]f[/math] is continuous over the closed interval [math][a,b][/math], and[/color][/*][*][color=#980000][math]f[/math] is differentiable over the open interval [math](a,b)[/math][/color], [/*][/list]then there exists at least one point [math]c[/math] in [math](a,b)[/math] for which [math]f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}[/math]. [br][br]Geometrically, the conclusion of the theorem says there exists a point [math]c[/math] in [math](a,b)[/math] for which the slope of the tangent line to [math]f[/math] at the point [math](c,f(c))[/math] is equal to the slope of the secant line through points [math](a,f(a))[/math] and [math](b,f(b))[/math].[br]

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]