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AHL 5.10—Second derivatives, testing for max and min - IB Questionbank | RevisionDojo

Practice AHL 5.10—Second derivatives, testing for max and min with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A function $f$ is defined for $-2 \leq x \leq 2$ , with the behavior of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ given below:

$x$	$-2<x<-1$	$x=-1$	$-1<x<1$	$1<x<2$
$f^{\prime}(x)$	Positive	0	Negative	Negative
$f^{\prime \prime}(x)$	Negative	0	Positive	Negative

Identify the x -coordinates of the local extrema of $f$ .

[3]

Determine the intervals where $f$ is concave up.

[2]

Given $f(-2)=3$ , sketch the graph of $f$ , indicating the local extrema and points of inflection.

[4]

Question 2

HLPaper 2

A particle moves in a straight line with displacement given by $s(t)=t^{4}-8 t^{3}+20 t^{2}-16 t+5$ , where $t \geq 0$ is time in seconds and $s$ is in meters.

Find expressions for the velocity $v(t)$ and acceleration $a(t)$ of the particle.

[3]

Determine the times when the particle is stationary.

[3]

Use the second derivative test to classify the nature of each stationary point.

[3]

Given that the particle's acceleration is zero at $t=p$ , find $p$ and determine whether this is a point of inflection.

[4]

Sketch the graph of $a(t)$ , indicating the point where $a(t)=0$ .

[3]

Question 3

HLPaper 1

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

Find $f^{\prime \prime}(x)$ .

[5]

Determine the points where the graph of $f$ is concave down.

[3]

Find the coordinates of any points of inflection and verify using the second derivative test.

[4]

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

[4]

Question 4

HLPaper 2

A particle moves along a straight line with displacement given by $s(t)=2 t^{3}-9 t^{2}+12 t+1$ , where $t \geq 0$ is time in seconds and $s$ is in meters.

Find the velocity $v(t)$ and acceleration $a(t)$ of the particle.

[3]

Determine the times when the particle is stationary.

[2]

Use the second derivative test to classify the nature of the stationary points of $s(t)$ .

[3]

Question 5

HLPaper 2

A particle moves along a curve defined by $y=x^{3} e^{-2 x}$ for $x \geq 0$ , where $x$ is time in seconds and $y$ is displacement in meters.

Find the velocity $v(x)=\frac{d y}{d x}$ .

[3]

Determine the exact coordinates of the stationary points of $y$ .

[4]

Sketch the graph of the acceleration $a(x)=\frac{d^{2} y}{d x^{2}}$ , indicating where $a(x)=0$ .

[2]

Question 6

HLPaper 1

Consider the function $f(x)=\frac{x^{2}+2}{x^{2}-4}$ for $x \neq \pm 2$ .

Find $f^{\prime}(x)$ .

[3]

Show that the graph of $f$ has no stationary points. [

[2]

Determine the intervals where $f(x)$ is concave down.

[2]

Question 7

HLPaper 1

Let $f(x)=x^{4}-4 x^{2}+2$ and $g(x)=\sin (2 x)$ for $x \in \mathbb{R}$ . Define $h(x)=f(x) \cdot g(x)$ .

Find $h^{\prime \prime}(x)$ .

[5]

Determine the x -coordinates of the stationary points of $h(x)$ in $[0, \pi]$ .

[4]

Use the second derivative test to classify the stationary point at $x=\frac{\pi}{4}$ .

[3]

Question 8

HLPaper 2

A camera is positioned 5 meters from a straight road. A car travels along the road at a constant speed of $20 \mathrm{~m} / \mathrm{s}$ . Let $\theta$ be the angle between the line from the camera to the point on the road closest to it and the line from the camera to the car.

Express $\theta$ in terms of the distance $x$ from the closest point to the car.

[2]

Find $\frac{d \theta}{d t}$ when the car is at the point closest to the camera.

[4]

Determine whether $\frac{d \theta}{d t}$ has a maximum or minimum at this point using the second derivative.

[4]

Question 9

HLPaper 1

Consider the function $f(x)=\frac{\ln \left(x^{2}+1\right)}{x}$ for $x \neq 0$ .

Find $f^{\prime}(x)$ , simplifying your answer.

[3]

Determine the intervals where $f(x)$ is increasing.

[3]

Show that $x=0$ is a point of inflection by evaluating the second derivative at $x=0$ .

[2]

Question 10

HLPaper 1

Consider the function $f(x)=x^{4}-8 x^{2}+3$ for $x \in \mathbb{R}$ .

Find the first derivative $f^{\prime}(x)$ .

[2]

Determine the coordinates of the stationary points of $f$ .

[3]

Use the second derivative test to determine the nature of each stationary point.

[3]

Sketch the graph of $f^{\prime}(x)$ , indicating the critical points.

[3]

AHL 5.10—Second derivatives, testing for max and min Questionbank