IB Math AI HL — Optimization

Key Idea: Optimisation is the real-world application of stationary points: finding the maximum profit, minimum cost, shortest time, or greatest area given a constraint. The technique is identical to Topic 5.6 — differentiate and set f'(x) = 0 — but now you must first build the function from a word problem, and interpret your answer with units in context.

✅ The 5-step optimisation method

Step 1: Define the variable

Choose one quantity as x. Be clear about what it represents and any constraints on its domain (e.g., x > 0, 0 < x < 10).

Step 2: Write the objective function

Express the quantity to maximise or minimise in terms of x. This is the function you will differentiate.

Step 3: Use the constraint

If there is a constraint (e.g., fixed perimeter, fixed volume), use it to eliminate one variable and express everything in terms of x.

Step 4: Differentiate and solve f'(x) = 0

Find the critical point. Solve the equation to get the x-value.

Step 5: Verify nature and interpret

Use f''(x) to confirm it is a max or min. Then find the actual maximum/minimum value and state it with units in context.

Example: A farmer has 100 m of fence to enclose a rectangular field with one side along a wall (no fence needed there). Maximise the area. Let x = width of the field. Then length = 100 − 2x. Area A = x(100 − 2x) = 100x − 2x² A'(x) = 100 − 4x = 0 → x = 25 m A''(x) = −4 < 0 → this is a maximum ✓ Length = 100 − 50 = 50 m. Maximum area = 25 × 50 = 1250 m²

Always check the domain after finding the critical x. An x that gives a negative length or area is outside the valid domain. If f''(x) = 0 at the critical point, test values either side of x to confirm max or min.

Paper 1 and 2: Show every step — setting up the function, differentiating, setting f'(x) = 0, verifying nature, and concluding in context. Marks are awarded at each stage. A correct final answer with no working earns few marks in extended questions. Context is essential: Don't just say 'x = 25' — say 'the width should be 25 m to achieve the maximum area of 1250 m²'.