Consider the function $f(x) = x^3 - 3x^2 - 9x + 2$ . Graph the function on your graphic display calculator and find the exact coordinates of the local maximum point. Find the exact coordinates of the local minimum point.

CG50: Go to MENU 5 (Graph) . Enter $Y1 = x^3 - 3x^2 - 9x + 2$ . DRAW . (M1) Press SHIFT -> F5 (G-Solv) -> F2 (MAX) . The calculator displays $X = -1, Y = 7$ . Coordinates: $(-1, 7)$ . A1 Press SHIFT -> F5 (G-Solv) -> F3 (MIN) . (M1) The calculator displays $X = 3, Y = -25$ . Coordinates: $(3, -25)$ . A1

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IB Maths AA SL · Unit 5: Calculus

IB Maths AA SL Graphical Behaviour and the Second Derivative Questions

Updated October 2026

Exam-style IB Maths AA SL graphical behaviour and the second derivative 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.

22 questions
Paper 1: 15
Paper 2: 7
4 easy
9 medium
6 hard
3 very hard
3 worked examples

Practise Graphical Behaviour and the Second Derivative 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

SL AA Paper 1 (non-calc) tests all three every year. Learn to spot which rule applies from the function structure — practise until you no longer 'reach for the formula'. Differentiation — product, quotient, chain rules overview →

Power rule reversed, plus the definite integral for area. SL AA Paper 1 expects you to know ∫e^x dx = e^x + c and ∫1/x dx = ln|x| + c by heart. Integration — antiderivatives and definite integrals overview →

What's examined in AA SL graphical behaviour and the second derivative

The question bank covers these graphical behaviour and the second derivative question types (number of questions in brackets):

Points of Inflection Calculation (12)
Concavity and Derivative Interpretation (5)
Local Extrema Classification (5)

Key formulas

Derivative of xⁿ: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Product rule: $(u v)^{'} = u^{'} v + u v^{'}$
Quotient rule: ${(\frac{u}{v})}^{'} = \frac{u^{'} v - u v^{'}}{v^{2}}$
Derivatives of standard functions: $\frac{d}{d x} (\sin x) = \cos x, \frac{d}{d x} (\cos x) = - \sin x$
Exponential & log derivatives: $\frac{d}{d x} (e^{x}) = e^{x}, \frac{d}{d x} (\ln x) = \frac{1}{x}$

In the same notation as the IB formula booklet. All AA SL formulas →

Graphical Behaviour and the Second Derivative worked examples

Worked example 1: Classifying a stationary point with $f^{″} (x)$ · easy

Let $f (x) = \frac{1}{3} x^{3} - 2 x^{2} - 21 x - 24$ . The graph of $f$ has a horizontal tangent at $x = - 3$ . Find $f^{″} (x)$ and prove whether the point at $x = - 3$ is a local maximum or local minimum.

Solution

6 steps · 1 to fill in

Examiner tip: Second-derivative test:

f^{″} (x) < 0

means maximum (frowning face),

f^{″} (x) > 0

means minimum (smiling face). If

f^{″} (x) = 0

the test is inconclusive and you must test the gradient just left and right.

Worked example 2: Non-stationary point of inflexion · medium

The curve $C$ has equation $f (x) = x^{3} - 6 x^{2} + 9 x - 1$ . Determine the $x$ -coordinate of the point of inflexion on the graph, and explain mathematically why this point of inflexion is NOT a stationary point.

Solution

6 steps · 1 to fill in

Examiner tip: A point of inflexion (change of concavity) is only a STATIONARY point of inflexion if the tangent at that exact point also happens to be horizontal (

f^{'} (x) = 0

Worked example 3: Proving a stationary point of inflexion · hard

A function is given by $f (x) = (x - 2)^{3} + 4$ . Show analytically that there is a stationary point of inflexion at $x = 2$ , clearly justifying your answer by analysing both the first and second derivatives.

Solution

6 steps

Examiner tip: Merely showing

f^{″} (x) = 0

is NOT sufficient to prove a point of inflexion. You must show the sign of the second derivative CHANGES across that point.

Try these IB Maths AA SL graphical behaviour and the second derivative 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 · easy · 4 marks · Paper 2

Consider the function $f (x) = x^{3} - 3 x^{2} - 9 x + 2$ .

Graph the function on your graphic display calculator and find the exact coordinates of the local maximum point.
Find the exact coordinates of the local minimum point.

Full worked solution Free, no sign-in

CG50: Go to MENU 5 (Graph). Enter $Y 1 = x^{3} - 3 x^{2} - 9 x + 2$ . DRAW. (M1)
Press SHIFT -> F5 (G-Solv) -> F2 (MAX).
The calculator displays $X = - 1, Y = 7$ .
Coordinates: $(- 1, 7)$ . A1
Press SHIFT -> F5 (G-Solv) -> F3 (MIN). (M1)
The calculator displays $X = 3, Y = - 25$ .
Coordinates: $(3, - 25)$ . A1

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

Question 2 · medium · 5 marks · Paper 2

A function $f$ is defined for $x > 0$ . The derivative of $f$ is given by: $f^{'} (x) = 3 x^{2} + \frac{2}{x^{3}} - 25$

Find $f^{″} (x)$ .
The graph of $f$ is concave up when $x > n$ , where $n$ is the least possible real number that makes this inequality true. Find the exact value of $n$ .

Attempt it and see the mark scheme →

Question 3 · hard · 6 marks · Paper 2

A function $f$ is defined for $x \neq 0$ . The derivative of $f$ is given by: $f^{'} (x) = 48 x^{2} + \frac{1}{x^{3}} - 22$

Find $f^{″} (x)$ .
Show that the curve $y = f (x)$ has exactly one point of inflexion, and determine its $x$ -coordinate.
Find the exact gradient of the normal line to the curve at this point of inflexion.

Attempt it and see the mark scheme →

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All 22 graphical behaviour and the second derivative questions with mark schemes →

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FAQ

How many IB Maths AA SL graphical behaviour and the second derivative questions are there?

There are 22 exam-style graphical behaviour and the second derivative questions in the AA SL question bank (Paper 1: 15 · Paper 2: 7), graded 4 easy, 9 medium, 6 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is graphical behaviour and the second derivative on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 15 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA 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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IB Maths AA SL Graphical Behaviour and the Second Derivative Questions

Try it first

What you need to know

What's examined in AA SL graphical behaviour and the second derivative

Key formulas

Graphical Behaviour and the Second Derivative worked examples

Worked example 1: Classifying a stationary point with $f^{″} (x)$ · easy

Solution

Worked example 2: Non-stationary point of inflexion · medium

Solution

Worked example 3: Proving a stationary point of inflexion · hard

Solution

Try these IB Maths AA SL graphical behaviour and the second derivative questions

Question 1 · easy · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 2 · medium · 5 marks · Paper 2

Question 3 · hard · 6 marks · Paper 2

Revise graphical behaviour and the second derivative for the exam

FAQ

What else is free here

More AA SL Unit 5 topics

IB Maths AA SL Graphical Behaviour and the Second Derivative Questions

Try it first

What you need to know

What's examined in AA SL graphical behaviour and the second derivative

Key formulas

Graphical Behaviour and the Second Derivative worked examples

Worked example 1: Classifying a stationary point with f″(x) · easy

Solution

Worked example 2: Non-stationary point of inflexion · medium

Solution

Worked example 3: Proving a stationary point of inflexion · hard

Solution

Try these IB Maths AA SL graphical behaviour and the second derivative questions

Question 1 · easy · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 2 · medium · 5 marks · Paper 2

Question 3 · hard · 6 marks · Paper 2

Revise graphical behaviour and the second derivative for the exam

FAQ

What else is free here

More AA SL Unit 5 topics

Worked example 1: Classifying a stationary point with $f^{″} (x)$ · easy