The [b]First Derivative Test for Local Extrema[/b] states that if [math]c[/math] is a critical point of a continuous function [math]f[/math], and if [math]f[/math] is differentiable at every point in some interval containing [math]c[/math] except possibly at [math]c[/math] itself, then[br][list=1][*]If [math]f'[/math] changes from [color=#cc0000]negative [/color]to [color=#0000ff]positive[/color] at [math]c[/math], then [math]f[/math] has a local minimum at [math]c[/math]. [br][/*][*]If [math]f'[/math] changes form [color=#0000ff]positive[/color] to [color=#cc0000]negative[/color] at [math]c[/math], then [math]f[/math] has a local maximum at [math]c[/math]. [br][/*][*]If [math]f'[/math] does not change sign at [math]c[/math], then [math]f[/math] has no local extremum at [math]c[/math]. [/*][/list]Drag the black point along the graph of function [math]f[/math]. As you do, its derivative and local extrema will appear. [br][br]Notice how [math]f[/math] has no local extremum at the first black point that appears on the function. [br]Why is this?

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