IB Mathematics Applications & Interpretation Topic AHL 5.10—second Derivatives, Testing for Max and Min Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.10—second Derivatives, Testing for Max and Min Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.10—second Derivatives, Testing for Max and Min with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.10—second Derivatives, Testing for Max and Min and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Consider the function $f (x) = \frac{x ^{3}}{x ^{2} + 4}$ for $x \in R$ .

Find $f^{''} (x)$ .

[3]

Determine the interval(s) of $x$ for which the graph of $f$ is concave down.

[2]

Find the coordinates of any points of inflection and verify by showing that $f^{''} (x)$ changes sign at each point.

[3]

Sketch the graph of $f^{''} (x)$ , indicating points where $f^{''} (x) = 0$ .

[2]

Question 2

HLPaper 1

Consider the function $g (x) = x e^{- x}$ for $x \in R$ .

Find $g^{'} (x)$ .

[3]

Find the $x$ -coordinate of the stationary point of $g$ .

[2]

Determine the interval where the graph of $g$ is concave down.

[3]

Question 3

HLPaper 1

For $x \in R$ , let $h (x) = \frac{x ^{2} - 1}{e ^{x} + 1}$ , where $e$ is the base of the natural logarithm.

Find $h^{'} (x)$ .

[3]

Using technology, solve $h^{'} (x) = 0$ to determine the $x$ -coordinates of the stationary points of $h (x)$ in $[- 1, 3]$ . Give your answers correct to 3 significant figures.

[3]

Use the second derivative test to classify the stationary point at $x \approx 2.55$ .

[3]

Question 4

HLPaper 2

A satellite moves along a straight path in space, and its displacement at time $t$ seconds is $s (t) = t^{3} - 6 t^{2} + 9 t$ metres, where $t \geq 0$ .

Calculate the velocity and acceleration functions of the satellite.

[3]

Determine the time(s) when the satellite is at rest.

[4]

Use the acceleration function to find the acceleration at $t = 2$ and state whether the satellite’s speed is increasing, decreasing, or momentarily constant at $t = 2$ .

[4]

Find the total distance travelled by the satellite from $t = 0$ to $t = 3$ .

[5]

Sketch the displacement-time graph for the satellite for $0 \leq t \leq 3$ , indicating the key features of its motion.

[4]

Question 5

HLPaper 2

A theatre sells $300$ tickets at $12 each for a concert. For every $1 increase in the ticket price, $10$ fewer tickets are expected to be sold. Let $x$ be the increase in price, in dollars.

Express the revenue $R$ as a function of $x$ .

[4]

Find the value of $x$ that maximizes the revenue.

[5]

Use the second derivative to confirm that the revenue is maximized at the value of $x$ found previously.

[4]

What is the maximum revenue?

[3]

IB Mathematics Applications & Interpretation AHL 5.10—second Derivatives, Testing for Max and Min Questionbank

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IB Mathematics Applications & Interpretation AHL 5.10—second Derivatives, Testing for Max and Min Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.10—second Derivatives, Testing for Max and Min Questionbank

Unlock all 37 questions with a free account