Curve Sketch with the First Derivative

Calculus 1 · Axiom Academy

Use the first derivative to locate critical points, intervals of increase/decrease, and the local extrema of a cubic. Analyze the function f(x) = x^3 - 3x + 2 . Find its critical points, determine where it is increasing and decreasing, and identify every local maximum and minimum. Rising into the peak at (-1, 4) , the curve falls to the valley at (1, 0) , then climbs again — exactly the increasing / decreasing / increasing pattern the sign chart predicted. The concavity flips at the inflection point (0, 2) . Nice work — you walked a cubic through the full first-derivative analysis. The procedure that got you there: First derivative: f'(x) = 3x^2 - 3 measures the slope, so it tells you where f rises and falls. Critical points: solving f'(x) = 0 gives x = -1 and x = 1 — the only candidates for local extrema. Sign analysis: means increasing and means decreasing; here the pattern is . First derivative test: f' flips at x=-1 (a local max) and at x=1 (a local min). Local extrema: local maximum at (-1, 4) and local minimum at (1, 0) . Slope sign locates the turning points and orders the increasing/decreasing intervals — the backbone of sketching any curve by hand.

This is the written version of the interactive lesson above. See the full Calculus 1 course.