This interactive figure illustrates the [b]first derivative theorem for local extreme values:[/b][list][*]If [math]f\left(x\right)[/math] has a local maximum or minimum value at an interior point [math]c[/math] of its domain, and if [math]f'[/math] is defined at [math]c[/math], then [math]f'(c) = 0[/math].[/*][/list]The [color=#980000][b]local extrema[/b][/color] at input values [i]x = c [/i]of the function's domain are labeled [i]A, B, C, D[/i], and [i]E[/i] (if they all exist). [br]The [b]New Function button [/b]will change the graph of the function. [br][br]Since the derivative of this function [i][math]f[/math][/i] is defined at each [color=#980000][b]local maximum[/b][/color] or [color=#980000][b]local minimum[/b][/color] point[br] [color=#980000][b][math](c,f(c))[/math][/b][/color] shown, we can conclude that [math]f'(c)=0[/math] at each of these [color=#980000][b]local extrema.[/b][/color]

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