Definition: A stationary point is any x-value where f′(x) = 0.
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.
Interactive graph: classify stationary points
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Method: find all stationary points
- Differentiate: find f′(x).
- Set f′(x) = 0 and solve for x.
- Find the y-coordinate: substitute each x into f(x).
- State the coordinates.
Example
Step by step
- f(x) = 2x³ − 9x² + 12x − 1
- Step 1 — f′(x) = 6x² − 18x + 12 = 6(x² − 3x + 2) = 6(x − 1)(x − 2)
- Step 2 — Set = 0: x = 1 or x = 2
- Step 3 — f(1) = 2 − 9 + 12 − 1 = 4. f(2) = 16 − 36 + 24 − 1 = 3.
- Result — Stationary points at (1, 4) and (2, 3).
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.
IB-style question — stationary points of a cubic [7 marks]
A function is defined by f(x) = 2x³ − 3x² − 36x + 5.
(a) Find the coordinates of the stationary points of f.
(b) Classify each stationary point.
(c) State the interval on which f is decreasing.
Step by step
- (a) Differentiate, then set the derivative equal to zero.
- Take out the common factor and factorise the quadratic.
- Substitute each x into f(x) for the y-coordinates.
- (b) Graph f on the GDC, or use a sign diagram of f′. The left point is a peak, the right point is a valley.
- (c) f is decreasing where f′(x) < 0, which is between the two stationary x-values.
Final answer
(a) (−2, 49) and (3, −76). (b) (−2, 49) is a local maximum; (3, −76) is a local minimum. (c) f is decreasing for −2 < x < 3.
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Classifying with the GDC
Two AI SL ways to classify: Once you have found a stationary point, decide whether it is a local maximum or minimum using EITHER the GDC graph OR a first-derivative sign diagram (next section).
In AI SL you do not use the second derivative.
The GDC method: Graph y = f(x) on the GDC.
A local maximum is a peak (the curve turns from rising to falling); a local minimum is a valley (falling to rising).
Use the GDC's maximum and minimum tools to read the coordinates directly.
Classifying with the GDC
Classify the stationary points of f(x) = 2x³ − 9x² + 12x − 1 (found at x = 1 and x = 2).
Step by step
- Graph y = 2x³ − 9x² + 12x − 1 on the GDC.
- The GDC maximum tool gives a peak at (1, 4).
- The GDC minimum tool gives a valley at (2, 3).
Final answer
Local maximum at (1, 4); local minimum at (2, 3).
GDC allowed on both papers: For a quick classification, graph the function and read which stationary point is the peak and which is the valley.
For an analytic justification, use the first-derivative sign diagram (next section).
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
Same example — sign diagram approach
Step by step
- Stationary points at x = 1 and x = 2
- Test x = 0 (region x < 1) — f′(0) = 12 > 0 → +
- Test x = 1.5 (region 1 < x < 2) — f′(1.5) = 6(2.25) − 27 + 12 = −1.5 < 0 → −
- Test x = 3 (region x > 2) — f′(3) = 54 − 54 + 12 = 12 > 0 → +
- Sign diagram — x: 1 2 f′: + | − | + max min
- Result — Local maximum at x = 1. Local minimum at x = 2.
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.