IB Maths AA SL · Unit 5: Calculus
IB Maths AA SL Graphical Behaviour and the Second Derivative Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 22 questions
- Paper 1: 15
- Paper 2: 7
- 4 easy
- 8 medium
- 6 hard
- 3 very hard
- 1 starter
- 3 worked examples
Practise Graphical Behaviour and the Second Derivative questions →
AA SL formula booklet
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 x^n
- \(\dfrac{d}{dx}\bigl(x^n\bigr) = n x^{\,n-1}\)
- Chain rule
- \(\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}\)
- Product rule
- \((uv)' = u'v + uv'\)
- Quotient rule
- \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
- Derivatives of standard functions
- \(\dfrac{d}{dx}(\sin x) = \cos x,\ \dfrac{d}{dx}(\cos x) = -\sin x,\ \dfrac{d}{dx}(\tan x) = \sec^2 x\)
- Exponential & log derivatives
- \(\dfrac{d}{dx}(e^x) = e^x,\ \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\)
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 $f(x) = \frac{1}{3}x^3 - 2x^2 - 21x - 24$. 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.
1. Differentiate: $f'(x) = x^2 - 4x - 21$.
2. Differentiate again: $f''(x) = 2x - 4$.
3. Substitute $x = -3$ into the second derivative: $f''(-3) = 2(-3) - 4$.
4. Evaluate: $f''(-3) = -6 - 4 = -10$.
5. Analyse the sign: $f''(-3) < 0$, so the curve is concave down here.
6. Conclude: the point at $x = -3$ is a $\mathbf{\text{local maximum}}$.
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 $f(x) = x^3 - 6x^2 + 9x - 1$. 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.
1. Find the first derivative: $f'(x) = 3x^2 - 12x + 9$.
2. Find the second derivative: $f''(x) = 6x - 12$.
3. Set $f''(x) = 0$: $6x - 12 = 0$.
4. Solve: $\mathbf{x = 2}$.
5. Check the gradient at $x = 2$: $f'(2) = 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3$.
6. Conclude: because $f'(2) = -3 \neq 0$ the tangent is not horizontal, so this is a $\mathbf{\text{non-stationary point of inflexion}}$.
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.
1. Find $f'(x)$ using the chain rule: $f'(x) = 3(x - 2)^2$.
2. Find $f''(x)$: $f''(x) = 6(x - 2)$.
3. Check stationary condition at $x = 2$: $f'(2) = 3(0)^2 = 0$. Tangent is horizontal.
4. Check inflexion condition at $x = 2$: $f''(2) = 6(0) = 0$. Potential change of concavity.
5. Verify concavity change by testing either side: $f''(1) = -6 < 0$ (concave down), $f''(3) = 6 > 0$ (concave up).
6. Conclude: since $f'(2) = 0$, $f''(2) = 0$, and the sign of $f''$ changes across $x = 2$, this is a $\mathbf{\text{stationary point of inflexion}}$.
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.
Question 1 · easy · 4 marks · Paper 2
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.
Question 2 · medium · 5 marks · Paper 2
A function \(f\) is defined for \(x > 0\). The derivative of \(f\) is given by:
\[f'(x) = 3x^2 + \frac{2}{x^3} - 25\]
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 \(n\).
Question 3 · hard · 6 marks · Paper 2
A function \(f\) is defined for \(x \neq 0\). The derivative of \(f\) is given by:
\[f'(x) = 48x^2 + \frac{1}{x^3} - 22\]
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.
All 22 graphical behaviour and the second derivative questions with mark schemes →
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, 8 medium, 6 hard, 3 very hard, 1 starter. 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.
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