Applications of Derivatives · Curve Analysis
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.
- 0 variables
- 2 worked examples
- 15 in Applications of Derivatives
Worked examples
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).
1 more worked examplePremium
Classify the critical points of f(x) = x³ - 6x² + 9x.
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