Relative Maxima and Minima

Pre-Calculus · Axiom Academy

LESSON Relative Maxima and Minima The local peaks and valleys of a graph — and the one thing that always happens right where the curve turns around. 1. Watch the Curve Turn Around Follow the point as it travels left to right along f(x) = x^3 - 3x . The trail is green while the function is increasing and red while it is decreasing . A relative maximum is the instant green hands off to red; a relative minimum is where red hands back to green. 2. Telling a Peak From a Valley Both a peak and a valley are points where the function turns around — but they turn in opposite directions. The tell is what the curve does just before and just after the point: The function is increasing ↗ — climbing up toward the peak. The function is decreasing ↘ — heading back down the other side. The function is decreasing ↘ — sliding down into the valley. The function is increasing ↗ — climbing back out. Peak / hilltop. Increasing → decreasing. f(c) is every nearby value. Valley / bowl bottom. Decreasing → increasing. f(c) is every nearby value. For f(x) = x^3 - 3x : the graph rises until x = -1 , giving a relative maximum value of f(-1) = 2 ; it falls until x = 1 , giving a relative minimum value of f(1) = -2 ; then it rises again. Two turns, one peak and one valley. 3. Why "Relative"? It's a Local Title

This is the written version of the interactive lesson above. See the full Pre-Calculus course.