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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.

  • 2 worked examples
  • 15 in Applications of Derivatives

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).

1 more worked examplePremium

Classify the critical points of f(x) = x³ - 6x² + 9x.

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Related formulas

Critical PointsSecond Derivative Test

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Common questions

What is the First Derivative Test?

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. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the First Derivative Test?

Worked example. Classify the critical points of f(x) = x³ - 3x. f'(x) = 3x² - 3 = 3(x-1)(x+1). Critical points: x = -1, x = 1. f'(-2) = 9 > 0, f'(0) = -3 < 0, f'(2) = 9 > 0. 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).

Practice

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