A stationary point is where the gradient is zero. To find it, solve dy/dx = 0 for x, then substitute into the curve for y. To decide the nature, use the second derivative.
This lesson is part of SPM Additional Mathematics differentiation. It uses the rules from differentiating powers, products and quotients.
The method in four steps
- Differentiate: find dy/dx.
- Solve dy/dx = 0 to get the x-values.
- Substitute each x into y to get the coordinates.
- Find d²y/dx² and test each x: negative means maximum, positive means minimum.
Worked example
Find the stationary points of y = x³ − 6x² + 9x + 2 and state their nature.
Gradient. dy/dx = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x − 1)(x − 3).
Stationary x-values. 3(x − 1)(x − 3) = 0 gives x = 1 or x = 3.
Coordinates. At x = 1: y = 1 − 6 + 9 + 2 = 6, so the point is (1, 6). At x = 3: y = 27 − 54 + 27 + 2 = 2, so the point is (3, 2).
Nature. d²y/dx² = 6x − 12. At x = 1 it is −6, which is negative, so (1, 6) is a maximum. At x = 3 it is 6, which is positive, so (3, 2) is a minimum.
Why the second derivative works
The second derivative tells you how the gradient is changing. A negative value means the gradient is falling, so it goes from positive through zero to negative, and the curve peaks. A positive value is the reverse, and the curve bottoms out.
The mistake that costs marks
The usual slip is to stop after finding x and leave the nature untested, or to state the nature without evidence.
| Step | Weak answer | Full answer |
|---|---|---|
| Point | x = 1 | (1, 6) |
| Nature | “It is a maximum” | d²y/dx² = −6 < 0, so maximum |
When d²y/dx² = 0, the test is silent. For y = x³ at x = 0, the gradient 3x² is positive on both sides, so the point is an inflexion and not a turning point.
Check yourself
Find the stationary points of y = 2x³ − 9x² + 12x − 1 and state their nature.
Answer
dy/dx = 6x² − 18x + 12 = 6(x − 1)(x − 2), so x = 1 or x = 2.
At x = 1: y = 2 − 9 + 12 − 1 = 4. At x = 2: y = 16 − 36 + 24 − 1 = 3.
d²y/dx² = 12x − 18. At x = 1 it is −6, so (1, 4) is a maximum. At x = 2 it is 6, so (2, 3) is a minimum.
What to study next
Stationary points lead straight into optimisation and rates of change. If a question limits x to an interval, read comparing endpoints and stationary points in a constrained optimisation.
To have a teacher read your working and tell you which line loses the mark, see online one-to-one Additional Mathematics tuition.