Definition: A stationary point is any x-value where f′(x) = 0.__LINEBREAK__At a stationary point, the tangent to the curve is horizontal. The function is momentarily neither increasing nor decreasing.
| Type | What it looks like | Sign of f′ either side |
|---|---|---|
| Local maximum | Curve peaks and turns down | + then − |
| Local minimum | Curve dips and turns up | − then + |
| Point of inflection | Curve flattens but keeps direction | + then + (or − then −) |
The word 'local' means it is the highest or lowest point in a nearby region — there may be higher or lower points elsewhere on the curve.
Stationary ≠ turning point: All turning points are stationary points. But not all stationary points are turning points — a point of inflection is stationary but does not change direction.
[Diagram: math-stationary-points] - Available in full study mode
Always find y-coordinates: Many students forget to substitute x back into f(x). The question asks for coordinates — both x and y are required.
Worked example
Apply the key method from Stationary Points and Their Nature in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
The second derivative test: The second derivative f″(x) is the derivative of f′(x).__LINEBREAK__At a stationary point where f′(a) = 0: • f″(a) < 0 → local maximum (curve bends downward) • f″(a) > 0 → local minimum (curve bends upward) • f″(a) = 0 → test is inconclusive — use a sign diagram instead
When the second derivative test fails: If f″(a) = 0, do not guess. Use a sign diagram of f′(x) to classify instead. The second derivative test is a shortcut — it doesn't always work.
Worked example
Apply the key method from Stationary Points and Their Nature in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
The sign diagram of f′(x) is the most reliable classification method — it always works, even when the second derivative test fails.
Classification rules from sign diagram: At a stationary point x = a: • f′ changes + → − : local maximum • f′ changes − → + : local minimum • f′ stays + → + (or − → −) : point of inflection
IB exam: both methods accepted: You can use either the second derivative or a sign diagram. Many students prefer the sign diagram because it shows the full picture.