IB Mathematics Analysis and Approaches Topic SL 5.7—the Second Derivative Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 5.7—the Second Derivative Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 5.7—the Second Derivative with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.7—the Second Derivative and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

SL & HLPaper 1

Let $g (x) = \frac{s i n ( 2 x )}{2} (\frac{1}{t a n ( x )} + 2) - sin^{2} (x)$ .

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

[5]

Show that $g^{''} (x) = - 4 g (x)$ .

[3]

Hence, using your answer from Part 2, find $\int g (x) d x$ in terms of $g^{'} (x)$ .

[4]

Question 2

SLPaper 1

Let $f (x) = x^{3} - 3 x^{2} + k x + 2$ , where $k$ is a constant. The function has a point of inflection at $x = 1$ and this point lies on the $x$ -axis.

Find the value of $k$ .

[4]

Determine the nature of the stationary points of $f (x)$ for this value of $k$ .

[4]

Question 3

SLPaper 2

A function $g$ is defined by $g (x) = x^{2} ln ∣ x ∣$ , where $x \neq = 0$ . The graph of $g$ has a point of inflection at $x = e^{a}$ .

Find the value of $a$ , and show that the second derivative changes sign at this point.

[5]

Sketch the graph of $g^{''} (x)$ for $x > 0$ , indicating the x-intercept and the behavior as $x \to 0^{+}$ and as $x \to \infty$ .

[3]

Find the area enclosed by the graph of $g^{''} (x)$ and the x-axis from $x = e^{a}$ to $x = e^{a + 1}$ .

[5]

Question 4

SLPaper 1

A function $f$ is defined by $f (x) = \frac{x ^{3}}{3} - k x^{2} + \frac{1}{x}$ , where $k \in R$ 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(s) of $k$ found in the previous part, find the intervals where the graph of $f$ is concave up. Use a sign diagram for $f^{''} (x)$ to justify your answer.

[4]

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

[4]

Question 5

SLPaper 1

Let $f (x) = \frac{1}{3} x^{3} - 2 x^{2} + 4 x - 1$ .

Find the second derivative of $f (x)$ and determine the $x$ -coordinate of the point of inflection.

[3]

Verify that $x = 2$ is a point of inflection by showing that $f^{''} (x)$ changes sign at $x = 2$ .

[4]

IB Mathematics Analysis and Approaches SL 5.7—the Second Derivative Questionbank

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IB Mathematics Analysis and Approaches SL 5.7—the Second Derivative Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.7—the Second Derivative Questionbank

Unlock all 20 questions with a free account