Using the First Derivative Test to Determine Relative Local Extrema

A relative extremum is a place where the graph turns around locally, a peak or a valley. At a peak the tangent line is tipping from pointing uphill to pointing downhill: f' goes from positive to negative. At a valley it's the reverse, f' goes from negative to positive. That sign flip is the whole idea. The derivative measures slope, and a turning point is exactly where slope quits being one sign and becomes the other.

To use this, first find the critical points: solve f'(x) = 0, and also note any x where f' is undefined but f is defined. Then build a sign chart. Pick a test value in each interval between consecutive critical points and check the sign of f' there. Reading left to right across a critical point, plus-to-minus means local max, minus-to-plus means local min, and no change means the graph just pauses or has an inflection-type flattening, not a turning point. For f(x) = x³ - 6x² + 9x + 2, f'(x) = 3x² - 12x + 9 = 3(x-1)(x-3). f' is positive for x<1, negative for 1<x<3, positive for x>3. So x=1 gives a local max, f(1)=6, and x=3 gives a local min, f(3)=2.

This test is the natural follow-up to finding critical points, and it sets up the Second Derivative Test, which uses concavity instead of a sign chart to do the same job faster when it applies. Together they're the main tools for locating and classifying the extrema that later get compared to endpoint values when hunting for absolute max and min on a closed interval.