Here we illustrate [b]Rolle's Theorem[/b]: if a function [math]f[/math] satisfies the three 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], [color=#980000]and[/color][/*][*][math]f\left(a\right)=f\left(b\right)[/math][br][/*][/list]then there exists at least one point [math]c[/math] in [math](a,b)[/math] for which [math]f'\left(c\right)=0[/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 tangent line to [math]f[/math] through the point [math](c,f(c))[/math] is horizontal.