IB Maths AI HL · Unit 5: Calculus
IB Maths AI HL The Second Derivative and Concavity Questions
Exam-style IB Maths AI HL the second derivative and concavity 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.
- 17 questions
- Paper 1: 17
- 5 easy
- 5 medium
- 5 hard
- 2 starter
- 3 worked examples
Practise The Second Derivative and Concavity questions →
AI HL formula booklet
What's examined in AI HL the second derivative and concavity
The question bank covers these the second derivative and concavity question types (number of questions in brackets):
- Inflexion Points & Test (8)
- Calculate Second Derivative (6)
- Concavity Intervals (3)
The Second Derivative and Concavity worked examples
Worked example 1: Finding the Second Derivative · easy
Consider the polynomial function $f(x) = 2x^4 - 3x^3 + 5x$. Find an expression for $f''(x)$ and calculate its exact value at $x = 1$.
1. Differentiate the function once to find $f'(x)$: $f'(x) = 8x^3 - 9x^2 + 5$.
2. Differentiate the first derivative to find the second derivative $f''(x)$: $f''(x) = 24x^2 - 18x$.
3. Substitute $x = 1$ into the second derivative equation.
4. Evaluate: $f''(1) = 24(1)^2 - 18(1) = 24 - 18$.
5. The exact value is $6$.
Examiner tip: The value of the second derivative tells you the concavity. Since $f''(1) = 6 > 0$, the curve is "smiling" (concave up) at the coordinate where $x=1$.
Worked example 2: Locating Points of Inflexion · medium
The curve $C$ is given by $y = x^3 - 6x^2 + 9x + 2$. Find the exact coordinates of the point of inflexion on curve $C$.
1. Find the first derivative: $\frac{dy}{dx} = 3x^2 - 12x + 9$.
2. Find the second derivative: $\frac{d^2y}{dx^2} = 6x - 12$.
3. Set the second derivative to zero to locate potential points of inflexion: $6x - 12 = 0 \implies 6x = 12 \implies x = 2$.
4. Substitute $x = 2$ back into the original curve equation to find the $y$-coordinate.
5. Evaluate: $y = (2)^3 - 6(2)^2 + 9(2) + 2 = 8 - 24 + 18 + 2 = 4$.
6. The exact coordinates of the point of inflexion are $(2, 4)$.
Examiner tip: While setting $f''(x)=0$ locates potential points of inflexion, a true point of inflexion only occurs if the concavity actually changes sign (from positive to negative, or vice versa) across that point.
Worked example 3: Applying the Second Derivative Test · hard
A function is defined as $f(x) = -2x^3 + 6x^2 - 5$. Find the $x$-coordinates of the stationary points and use the second derivative test to classify their nature.
1. Find the first derivative: $f'(x) = -6x^2 + 12x$.
2. Set $f'(x) = 0$ to find stationary points: $-6x(x - 2) = 0 \implies x = 0$ and $x = 2$.
3. Find the second derivative: $f''(x) = -12x + 12$.
4. Test $x = 0$ using the second derivative: $f''(0) = -12(0) + 12 = 12$. Since $12 > 0$, the curve is concave up, so $x=0$ is a local minimum.
5. Test $x = 2$ using the second derivative: $f''(2) = -12(2) + 12 = -12$. Since $-12 < 0$, the curve is concave down, so $x=2$ is a local maximum.
Examiner tip: If the second derivative test results in $0$, the test is inconclusive. You must then use the first derivative test (checking gradients slightly to the left and right of the point) to classify the stationary point.
Try these IB Maths AI HL the second derivative and concavity 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 1
Given the function \(f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7\), find an expression for the second derivative, \(f''(x)\).
Question 2 · medium · 5 marks · Paper 1
Consider the function \(f(x) = x^4 - 6x^2 + 4\).
Determine the exact intervals for \(x\) where the curve is concave down.
Question 3 · hard · 7 marks · Paper 1
Consider the function \(f(x) = x e^{-x}\).
Use the product rule to find \(f'(x)\) and \(f''(x)\), and hence prove algebraically that the curve has exactly one point of inflexion at \(x = 2\).
All 17 the second derivative and concavity questions with mark schemes →
FAQ
How many IB Maths AI HL the second derivative and concavity questions are there?
There are 17 exam-style the second derivative and concavity questions in the AI HL question bank (Paper 1: 17), graded 5 easy, 5 medium, 5 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is the second derivative and concavity on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL 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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