Stationary points
Objective. Locate and classify turning points using derivative evidence.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
A stationary point occurs where the gradient of a curve equals zero, found by solving f'(x) = 0, and includes local maxima, local minima and points of inflection. To classify a stationary point, use the second derivative: if f''(x) > 0 it is a minimum, if f''(x) < 0 it is a maximum, and if f''(x) = 0 further investigation is needed using the gradient on either side. For example, for f(x) = x³ − 3x, f'(x) = 3x² − 3 = 0 gives x = ±1, and f''(x) = 6x shows x = 1 is a minimum and x = −1 is a maximum. These points reveal the shape and turning behaviour of the curve.
Worked example
Find and classify the stationary points of f(x) = x³ − 12x.
- 1Differentiate: f'(x) = 3x² − 12, set equal to 0: x² = 4, so x = ±2.
- 2Find f''(x) = 6x.
- 3At x = 2, f''(2) = 12 > 0, so it is a minimum; at x = −2, f''(−2) = −12 < 0, so it is a maximum.
- 4Compute y-values: f(2) = 8 − 24 = −16; f(−2) = −8 + 24 = 16.
Answer: Minimum (2, −16); maximum (−2, 16)
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Start practisingCommon mistake. Students often stop after finding f'(x) = 0 without substituting back to find the corresponding y-coordinate or without checking the second derivative, so the stationary point remains unclassified and incomplete.
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