Quick answer: Write the quantity to optimise in terms of two variables, use the constraint to remove one, differentiate, and solve f′(x) = 0. Then justify the type of point with the second derivative or a sign change of f′(x), and answer in context with units.
Maximise a profit or minimise a surface area: set up the function from the context, use the constraint to reduce it to one variable, solve f′(x) = 0 and justify that you have a maximum or a minimum.
Practise Optimisation in context with AI marking
Open the full HL AI engine — every question in this unit is worked through with instant AI-graded feedback on your handwritten workings.
Name the variables and draw a diagram; most of the marks are in setting up the function.
The constraint (a fixed volume, perimeter or budget) lets you write one variable in terms of the other.
Justify: f″(x) > 0 at the point means a minimum and f″(x) < 0 a maximum. Or show that f′(x) changes sign.
Check the end points of the domain too: on a closed interval the greatest value can be at an end.
The GDC can find the maximum or minimum of the function's graph as a check, but a "show that" or "justify" part needs the working.
Worked example
A closed cylindrical can must hold 500 cm³. Find the radius that minimises the surface area, justify that it is a minimum, and give the least surface area.
Volume: πr²h = 500, so h = 500 ÷ (πr²).
Surface area: S = 2πr² + 2πrh = 2πr² + 1000/r.
S′(r) = 4πr − 1000/r² = 0 gives r³ = 250/π, so r ≈ 4.30 cm (and h ≈ 8.60 cm, twice the radius).
S″(r) = 4π + 2000/r³ ≈ 37.7 > 0 at r ≈ 4.30, so this is a minimum.
S ≈ 2π(4.30)² + 1000 ÷ 4.30 ≈ 349 cm².
Answer: r ≈ 4.30 cm (h ≈ 8.60 cm); the least surface area is about 349 cm².
How do you show that a stationary point is a maximum or a minimum?
Use the second derivative: if f″(x) is negative at the point it is a maximum, and if positive a minimum. If f″(x) is zero or hard to find, show that f′(x) changes from positive to negative (maximum) or from negative to positive (minimum).
Can I use my calculator for optimisation in context?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working.