The big idea: A local maximum is a peak — the highest point in its neighbourhood.
A local minimum is a valley — the lowest point in its neighbourhood.
Together they are called turning points.
A turning point is where the graph changes direction: it stops going up and starts going down (maximum), or stops going down and starts going up (minimum).
- Local maximum
- A point (a, f(a)) where f(a) is greater than all nearby values. The graph peaks here.
- Local minimum
- A point (a, f(a)) where f(a) is less than all nearby values. The graph valleys here.
- Turning point
- Any local maximum or minimum — where the graph changes direction.
Local maximum
- Peak — highest point in its neighbourhood
- Graph changes from rising to falling
- Shape: ∩ (upside-down U near the point)
- Example: highest profit before costs rise
Local minimum
- Valley — lowest point in its neighbourhood
- Graph changes from falling to rising
- Shape: ∪ (U-shape near the point)
- Example: lowest temperature before warming
Local vs global: A local maximum is the highest point in its immediate area — not necessarily the highest point overall.
IB questions usually ask for local turning points, so focus on what you can see in the given domain.
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The big idea: On a graph, you can read the coordinates of any turning point directly.
The x-coordinate tells you where the turning point is; the y-coordinate tells you the maximum or minimum value.
Reading a turning point
A graph of y = f(x) has a peak at the point (3, 7).
State the local maximum value and where it occurs.
Step by step
- Identify the coordinates of the peak.
- The x-coordinate tells you the location.
- The y-coordinate is the maximum value.
Final answer
Local maximum of 7 at x = 3.
| Feature | Local Maximum | Local Minimum |
|---|---|---|
| Shape at the point | Peak (∩ shape) | Valley (∪ shape) |
| y-value compared to neighbours | Higher than nearby points | Lower than nearby points |
| GDC label | Maximum | Minimum |
How to write your answer: IB expects: 'The local maximum value is [y] and it occurs at x = [x].' Give both coordinates — just the y-value or just the x-value is incomplete.
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The big idea: you find turning points using your GDC — not calculus.
Enter the function, graph it, then use the built-in maximum/minimum finder to get exact coordinates.
GDC steps (any model)
- Enter f(x) into Y₁ (TI) or the graph editor (Casio / Nspire).
- Choose a window that includes the turning point you are looking for.
- Open the calculator's maximum / minimum tool: TI-84 — CALC → 4:maximum or 3:minimum; Casio fx-CG — G-Solv → MAX or MIN; TI-Nspire — Menu → Analyze Graph → Maximum or Minimum.
- When prompted, give a left bound to the LEFT of the turning point, a right bound to the RIGHT, then a guess close to the turning point.
- Read both coordinates from the screen — write them to 3 s.f. unless told otherwise.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Accuracy on GDC answers: IB awards marks for the correct coordinates read from your GDC.
Always write both x and y.
Give 3 significant figures unless the question specifies exact values.
If you only state 'maximum ≈ 5', you may lose a mark for missing the x-coordinate.
Set a sensible window: If the question gives a domain, restrict your GDC window to that domain.
A turning point outside the domain should not be reported.
IB-style question — maximum in context [4 marks]
A shop's daily revenue, R thousand dollars, when it sells x hundred items is modelled by R(x) = −3x² + 48x − 60, for 0 ≤ x ≤ 16.
(a) Use your GDC to find the number of items that maximises the daily revenue.
(b) Find the maximum daily revenue.
Step by step
- Graph R(x) on the GDC for 0 ≤ x ≤ 16 and use the maximum tool to read the peak.
- (a) x is measured in hundreds of items, so the maximising sales level is 8 × 100.
- (b) The y-coordinate of the peak is the maximum revenue, in thousands of dollars.
- Read the peak straight off the GDC and convert each coordinate back into the words of the question (items and dollars).
Final answer
(a) 800 items. (b) Maximum daily revenue is $132 000.
The big idea: When a problem describes a real situation, a local maximum or minimum often has a meaningful interpretation.
Always connect the coordinates back to the context — say what x and y represent.
Contextual turning point
The height of a ball in metres is modelled by h(t) = −5t² + 20t + 2, where t is time in seconds.
Find the maximum height and the time at which it occurs.
Step by step
- Graph h(t) on the GDC. Use Analyze Graph → Maximum.
- Read the coordinates of the peak.
- Interpret in context.
Final answer
The ball reaches a maximum height of 22 m after 2 seconds.
Minimum running cost — production model
A factory's daily cost (in thousands of dollars) to produce x widgets is C(x) = 0.5x² − 8x + 50.
Find the production level that minimises the cost and state that minimum cost.
Step by step
- Graph C(x) on the GDC. Use CALC → 3:minimum (TI-84) or Analyze → Minimum (Nspire).
- Set left bound around x = 4, right bound around x = 12, guess near the trough.
- Read the coordinates of the minimum.
Final answer
The minimum cost is $18,000 at a production level of 8 widgets. Below or above 8 widgets the cost is higher.
Context interpretation answers: IB usually asks: 'State the maximum value and when it occurs.' Your answer must include units and a sentence linking back to the context (height, profit, temperature, etc.).
A bare number with no context earns partial credit at best.