CalcRef

Applications of Derivatives · Curve Analysis

First Derivative Test

f′ changes +→−⇒local max;f′ changes −→+⇒local minf' \text{ changes } + \to - \Rightarrow \text{local max}; \quad f' \text{ changes } - \to + \Rightarrow \text{local min}

At a critical point c: if f' changes from positive to negative, c is a local max. If f' changes from negative to positive, c is a local min.

Worked examples

Classify the critical points of f(x) = x³ - 3x.
  1. f'(x) = 3x² - 3 = 3(x-1)(x+1). Critical points: x = -1, x = 1
  2. f'(-2) = 9 > 0, f'(0) = -3 < 0, f'(2) = 9 > 0
  3. At x = -1: f' changes + to - → local max. At x = 1: f' changes - to + → local min

Answer: Local max at x = -1 (f = 2), local min at x = 1 (f = -2).

Related formulas

Critical PointsSecond Derivative Test

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