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SL 5.7—The second derivative - IB Questionbank | RevisionDojo

Practice SL 5.7—The second derivative with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Consider the function $f(x) = 3x^4 - 8x^3 + 6x^2 - 4x + 1$ .

Find the first derivative $f'(x)$ .

[2]

Find the second derivative $f''(x)$ .

[2]

Find the points of inflection of the function.

[3]

Question 2

SL & HLPaper 1

Consider the function $h(x) = x^3 - 6x^2 + 9x + 1$ .

Determine the critical points and classify them as local maxima, minima, or points of inflexion.

[6]

Question 3

SL & HLPaper 1

Consider the function $g(x)=\frac{\sin(2x)}{2}(\frac{1}{\tan(x)}+2)-\sin^2(x)$

Show that $g(x)=\sin(2x)+\cos(2x)$

[5]

Show that $g''(x)=-4g(x)$

[3]

Hence, using your answer from part (b), find $\int\!g(x)\,dx$ in terms of $g(x)$

[4]

Question 4

HLPaper 1

In this question, all lengths are in metres and time is in seconds. Consider two particles, $A_1$ and $A_2$ , which start to move at the same time. Particle $A_1$ moves in a straight line such that its displacement from a fixed-point is given by $s(t) = 10 - \frac{7}{4}t^2$ , for $t \geq 0$ .

Find an expression for the velocity of ${A}_1$ at time ${t}$ .

[2]

Particle $A_2$ also moves in a straight line. The position of $A_2$ is given by $r=-\frac{1}{6}+t\left(\begin{array}{c}4\\-3\end{array}\right).$ The speed of $A_1$ is greater than the speed of $A_2$ when $t>q$ . Find the value of $q$ .

[5]

Question 5

SLPaper 1

A function $g$ is defined by $g(x)=x^{3}-6 x^{2}+12 x-4$ . The second derivative of $g$ is given by $g^{\prime \prime}(x)=6 x-6$ .

Find the intervals where the graph of $g$ is concave up and concave down.

[3]

Sketch the graph of $g^{\prime \prime}(x)$ , clearly indicating the x-intercept and y-intercept. marks]

[3]

Determine the nature of the stationary points of $g(x)$ using the second derivative test.

[3]

Question 6

SLPaper 1

Consider the function $f(x)=2 x^{3}-9 x^{2}+12 x-3$ .

Find the coordinates of the point of inflection of $f(x)$ .

[3]

Sketch the graph of $f(x)$ , showing the point of inflection and the stationary points.

[3]

Question 7

SLPaper 1

A function $f$ is defined by $f(x)=\frac{1}{4} x^{4}-2 x^{3}+q x^{2}+5$ , where $q \in \mathbb{R}$ . The graph of $f$ has a point of inflection at $x=2$ .

Find the value of $q$ .

[4]

Find the coordinates of the point of inflection and verify that it is indeed a point of inflection by examining the change in concavity.

[3]

Question 8

SLPaper 1

A function $g$ is defined by $g(x)=x^{4}-8 x^{2}+p x+5$ , where $p \in \mathbb{R}$ . The graph of $g$ has a point of inflection at $x=0$ .

Find the value of $p$ .

[3]

Sketch the graph of $g^{\prime \prime}(x)$ , clearly indicating the x-intercepts and y-intercept. marks]

[2]

Determine the nature of the stationary points of $g$ using the second derivative test.

[3]

Question 9

SLPaper 2

A particle moves along a straight line with velocity $v(t)=t^{2} e^{-t}$ , where $t \geq 0$ is time in seconds and $v$ is in $\mathrm{m} / \mathrm{s}$ .

Show that the acceleration of the particle is given by $a(t)=t e^{-t}(2-t)$ , and find the times when the acceleration is zero.

[4]

Prove that the position-time graph has exactly one point of inflection, and find its coordinates.

[5]

Find the maximum velocity of the particle, and determine whether the positiontime graph is concave up or concave down at this point. marks]

[4]

Question 10

SLPaper 1

A function $f$ is defined by $f(x)=\frac{x^{3}}{3}-k x^{2}+\frac{1}{x}$ , where $k \in \mathbb{R}$ and $x \neq 0$ . The graph of $f$ has exactly one point of inflection.

Find the value of $k$ such that the graph of $f$ has exactly one point of inflection, and determine the x -coordinate of this point.

[5]

For the value of $k$ found in part (a), find the intervals where the graph of $f$ is concave up, and justify your answer using a sign diagram for $f^{\prime \prime}(x)$ .

[4]

Determine the nature of the stationary points of $f$ using the second derivative test.

[4]

SL 5.7—The second derivative Questionbank