First Derivative: Increasing/Decreasing, Max/Min #
- \(f’(x) > 0\) → function increasing ↗
- \(f’(x) < 0\) → function decreasing ↘
- \(f’(x) = 0\) → stationary point
| Before | After | Type |
|---|---|---|
| + (increasing) | - (decreasing) | Relative maximum |
| - (decreasing) | + (increasing) | Relative minimum |
Example: \(f(x) = x^3 - 3x\) #
\(f’(x) = 3x^2 - 3 = 3(x-1)(x+1)\). Stationary points: \(x = -1\) (max at \((-1, 2)\)) and \(x = 1\) (min at \((1, -2)\)).
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