At a stationary point the gradient is zero, so solve dy/dx = 0. For y = x³ + 6x² + 9x + 1, dy/dx = 3x² + 12x + 9 = 3(x + 1)(x + 3), so the stationary points are at x = −1 and x = −3.
Use the second derivative to find their nature. If d²y/dx² > 0 the point is a minimum; if it is < 0, a maximum. Here d²y/dx² = 6x + 12: at x = −1 it is 6, a minimum, and at x = −3 it is −6, a maximum. Substitute into y to get the coordinates.
Worked example
Find the stationary point of y = 2x² and its nature.
- dy/dx = 4x = 0, so x = 0 and y = 0.
- d²y/dx² = 4 > 0.
Answer:(0, 0), a minimum
Key words
- stationary point
- where dy/dx = 0
- maximum
- a top of the curve
- minimum
- a bottom of the curve
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