$A(r) = 2\pi r^2 + \frac{2000\pi}{r}$. Find the radius (3 s.f.) that minimises surface area.

1. Recognise : fastest via GDC graph. 2. Enter : $Y_1 = 2\pi x^2 + \frac{2000\pi}{x}$. 3. Window : $x > 0$. 4. Locate minimum : G-Solv $\to$ MIN. 5. Read $x$ : $7.9370\ldots$ 6. State : $\mathbf{r = 7.94\text{ cm}}$.

The volume of a weather balloon during a test, $V(t)$, over a 4-hour period is modelled by $V(t) = 2t^3 - 12t^2 + 18t + 5$ for $0 \le t \le 4$. Find the times $t$ at which the volume has a local maximum or local minimum.

Differentiate the volume: $V'(t) = 6t^2 - 24t + 18$. A1 Set the derivative equal to zero: $6(t^2 - 4t + 3) = 0$. M1 Factorise: $(t-1)(t-3) = 0 \implies t = 1, \ t = 3$. A1 Check the sign of $V'(t)$ either side: $V'(0) = 18 > 0$, $V'(2) = -6 0$, so the gradient changes from positive to negative at $t = 1$ (local maximum) and from negative to positive at $t = 3$ (local minimum). R1

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IB Maths AI SL · Unit 5: Calculus

IB Maths AI SL Optimisation Questions

Updated October 2026

Exam-style IB Maths AI SL optimisation questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

15 questions
Paper 1: 5
Paper 2: 10
4 medium
5 hard
5 very hard
1 starter
3 worked examples

Practise Optimisation questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

1) Write down what to maximise/minimise. 2) Express in one variable. 3) Differentiate + set = 0. 4) Verify it's a max not a min. Every SL AI optimisation question follows this frame. Optimisation — the 4-step method overview →

Optimisation worked examples

Worked example 1: Maximising a quadratic · easy

$P (x) = - 2 x^{2} + 120 x - 500$ . Find the exact number of components that maximises daily profit.

Solution

4 steps

Examiner tip: Using calculus (set

P^{'} (x) = 0

) guarantees full method marks on Unit 5 optimisation, even for a quadratic.

Worked example 2: Maximising volume of an open box · medium

$V (x) = 27 x - 0.75 x^{3}$ (cm $^{3}$ ). Find the exact positive $x$ that maximises the volume.

Solution

5 steps · 1 to fill in

Examiner tip: Verify calculus answers using the MAX tool on the GDC's graph screen — a great safety net against sign or decimal errors.

Worked example 3: Minimising surface area using GDC · hard

$A (r) = 2 π r^{2} + \frac{2000 π}{r}$ . Find the radius (3 s.f.) that minimises surface area.

Solution

6 steps · 1 to fill in

Examiner tip: When a consolidated function is provided and calculus isn't explicitly demanded, jump straight to the GDC graph minimum tool.

Try these IB Maths AI SL optimisation questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · medium · 4 marks · Paper 1

The volume of a weather balloon during a test, $V (t)$ , over a 4-hour period is modelled by $V (t) = 2 t^{3} - 12 t^{2} + 18 t + 5$ for $0 \leq t \leq 4$ . Find the times $t$ at which the volume has a local maximum or local minimum.

Full worked solution Free, no sign-in

Differentiate the volume: $V^{'} (t) = 6 t^{2} - 24 t + 18$ . A1
Set the derivative equal to zero: $6 (t^{2} - 4 t + 3) = 0$ . M1
Factorise: $(t - 1) (t - 3) = 0 ⟹ t = 1, t = 3$ . A1
Check the sign of $V^{'} (t)$ either side: $V^{'} (0) = 18 > 0$ , $V^{'} (2) = - 6 < 0$ , $V^{'} (4) = 18 > 0$ , so the gradient changes from positive to negative at $t = 1$ (local maximum) and from negative to positive at $t = 3$ (local minimum). R1

Every question here has a full worked solution. See them all →

Question 2 · hard · 7 marks · Paper 2

A company manufactures and sells $x$ electronics units per week. Their weekly cost function, in euros, is given by $C (x) = 0.5 x^{2} - 10 x + 200$ . The revenue from selling $x$ units is modelled by $R (x) = 40 x$ .

Formulate an expression for the weekly profit function, $P (x)$ .
Find the derivative of the profit function, $P^{'} (x)$ .
Use calculus to determine the number of units the company should produce to maximize their weekly profit, and calculate this maximum profit.

Attempt it and see the mark scheme →

Question 3 · very hard · 7 marks · Paper 2

A cylindrical tin can must have a fixed volume of $500 {cm}^{3}$ . The cost of the metal for the top and bottom circles is $3 {cents per cm}^{2}$ , and the cost of the metal for the curved side is $2 {cents per cm}^{2}$ .

Let $r$ be the radius of the base. Show that the total cost of the can is given by $C (r) = 6 π r^{2} + \frac{2000}{r}$ .
Use differential calculus to find the exact value of $r$ that minimizes the total cost.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL optimisation questions are there?

There are 15 exam-style optimisation questions in the AI SL question bank (Paper 1: 5 · Paper 2: 10), graded 5 hard, 5 very hard, 4 medium, 1 starter. Every question has a full IB-style mark scheme (M, A and R marks).

Is optimisation on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 5 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lesson (slides and teacher notes): Optimisation problems.

IB Maths AI SL Optimisation Questions

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What you need to know

Optimisation worked examples

Worked example 1: Maximising a quadratic · easy

Solution

Worked example 2: Maximising volume of an open box · medium

Solution

Worked example 3: Minimising surface area using GDC · hard

Solution

Try these IB Maths AI SL optimisation questions

Question 1 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 2 · hard · 7 marks · Paper 2

Question 3 · very hard · 7 marks · Paper 2

Revise optimisation for the exam

FAQ

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