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.
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: For AI SL, 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 (TI-Nspire / Casio fx-CG)
- Enter f(x) in the graph application
- Graph the function over the required domain
- Use Analyze Graph → Maximum or Minimum
- Set left and right bounds around the turning point
- Read the coordinates from the screen — write them to 3 s.f. unless told otherwise
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.
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.
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.