Core idea: Optimisation means finding the maximum or minimum value of a quantity.__LINEBREAK__Examples: • Maximum profit • Minimum cost • Maximum area • Minimum material used__LINEBREAK__The method: the optimal value always occurs at a stationary point of the model function.
Every optimisation problem gives you a formula (or you build one), and asks you to find the input value that makes the output as large or as small as possible.
| Goal | What you find | IB command terms |
|---|---|---|
| Maximum | Largest value of the function | maximise, greatest, most |
| Minimum | Smallest value of the function | minimise, least, cheapest |
Same maths, different context: Whether the question involves profit, area, time, or temperature — the calculus steps are identical. Only the interpretation changes.
Always state the optimal value: Finding x = 4 is not the final answer. You must substitute into the original function to find the maximum or minimum value itself, and state the units.
Worked example
Apply the key method from Optimisation in Context 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.
Never wonder what to study next
Get a personalized daily plan based on your exam date, progress, and weak areas. We'll tell you exactly what to review each day.
IB exam questions frequently ask you to justify whether the stationary point is a maximum or minimum. Simply saying 'because x = …' earns no marks.
Two methods — both acceptable in IB: Method 1: Second derivative If f″(a) < 0 → maximum. If f″(a) > 0 → minimum.__LINEBREAK___Method 2: Sign diagram of f′ If f′ changes from + to − → maximum. From − to + → minimum.
Don't forget domain constraints: If the problem gives a restricted domain (e.g. 0 ≤ x ≤ 10), also check the endpoints. The global maximum or minimum might be at an endpoint, not at the stationary point.
Worked example
Apply the key method from Optimisation in Context 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.
Some IB questions give you the formula directly. Others require you to build it from a word problem. This is the harder skill — but it follows a consistent pattern.
Constraint elimination: When you have two variables, the constraint (total perimeter, total cost, etc.) lets you write one variable in terms of the other — reducing to a single-variable function you can differentiate.