Stationary points
A stationary point has derivative zero, but it need not be a maximum or minimum.
Solve f′(x) = 0 and substitute each solution into f(x). Then classify the point. A positive second derivative gives a local minimum, a negative second derivative gives a local maximum. A zero second derivative is inconclusive.
A sign change in f′ from negative to positive indicates a minimum; positive to negative indicates a maximum. A stationary inflection can have no sign change. For a closed-interval optimisation task, compare stationary values with endpoint values.
Worked example
Classify the stationary point of y = x² − 6x + 8.
- dy/dx = 2x − 6, which is zero at x = 3.
- At x = 3, y = −1.
- The second derivative is 2 > 0.
Answer: A minimum at (3,−1)
Revise first: Differentiating powers.
Practise stationary points
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 4.6: Second derivatives
- 8365 · core · 4.7: Classifying stationary points
- 8365 · core · 4.7: A polynomial stationary point
- 9FM0 · core · Assumed A-level Mathematics knowledge: A polynomial stationary point
- 7367 · core · Assumed A-level Mathematics knowledge: A polynomial stationary point
- H245 · core · Assumed A-level Mathematics knowledge: A polynomial stationary point
- H645 · core · Assumed A-level Mathematics knowledge: A polynomial stationary point
- 6993 · core · CA5–CA6: A polynomial stationary point
- 4PM1 · core · 9D–9E: A polynomial stationary point
Next practice: Sketching functions.