IB Mathematics Analysis and Approaches Topic SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 1

A rectangular garden is to be enclosed by a fence with a fixed perimeter of $40$ metres. Let the length of the garden be $x$ metres and the width be $y$ metres.

Express the area $A$ of the garden as a function of $x$ .

[2]

Find the value of $x$ that maximises $A$ . Verify that this gives a maximum using the second derivative test.

[4]

Calculate the maximum area of the garden.

[1]

Question 2

SLPaper 1

A right circular cylinder of radius $r$ and height $h$ is inscribed in a hemisphere of fixed radius $R$ , such that the base of the cylinder lies on the base of the hemisphere.

Show that the volume $V$ of the cylinder can be expressed as $V = π h (R^{2} - h^{2})$ .

[2]

Find the ratio of the height $h$ to the radius $R$ that maximizes the volume of the cylinder.

[4]

Hence, find the exact maximum volume in terms of $R$ .

[2]

Question 3

SLPaper 1

The graph of the derivative $f^{'} (x) = 2 x^{2} - 8 x + 6$ of a function $f$ is defined for all real numbers. The region enclosed by the graph of $f^{'} (x)$ and the $x$ -axis from $x = 1$ to $x = 3$ has an area of 4 units $^{2}$ .

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

[2]

Determine the $x$ -coordinates of the local maximum and minimum points of $f$ . Justify using the first derivative test.

[3]

Find the $x$ -coordinate of the point of inflection of $f$ .

[2]

Given that $f (3) = 5$ , find $f (x)$ .

[6]

Question 4

SLPaper 2

A population of bacteria is modeled by the differential equation $\frac{d P}{d t} = \frac{P ( k - P )}{k t}$ , where $P (t)$ is the population at time $t \geq 1$ seconds, $k > 0$ is the carrying capacity, and $P (1) = 500$ . The population triples by $t = 5$ , i.e., $P (5) = 1500$ .

Use the substitution $P = k v$ to show that $\frac{d v}{v ( 1 - v )} = \frac{d t}{t}$ .

[4]

Solve the differential equation to find $P (t)$ in terms of $k$ and $t$ .

[8]

Find the value of $k$ , correct to four significant figures.

[3]

Determine the time $t$ when the rate of change of the population is maximized for $t \geq 1$ , and state this maximum rate, correct to three significant figures.

[5]

Sketch the graph of the rate of change $\frac{d P}{d t}$ for $t \geq 1$ , indicating the maximum rate and its corresponding time.

[3]

Question 5

SLPaper 2

Consider the function $f (x) = \frac{l n x}{x ^{2} + x + 1}$ for $x > 0$ .

Show that the first derivative of $f$ is $f^{'} (x) = \frac{x ^{2} + x + 1 - x ( 2 x + 1 ) l n x}{x ( x ^{2} + x + 1 ) ^{2}}$ .

[4]

Find the coordinates of the point where the graph of $f$ has a horizontal tangent. Give your answer correct to $3$ decimal places.

[6]

Given that the second derivative is $f^{''} (x) = \frac{( 2 x + 1 ) ( ( 2 x + 1 ) l n x - 2 )}{x ( x ^{2} + x + 1 ) ^{3}}$ , determine the $x$ coordinate of the point of inflection in the interval $0 < x < 1$ . Justify your answer.

[6]

Sketch the graph of $f$ for $x > 0$ , labelling the horizontal tangent point, point of inflection in $0 < x < 1$ , the $x$ -intercept, and the asymptotic behaviour.

[3]

IB Mathematics Analysis and Approaches SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion Questionbank

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IB Mathematics Analysis and Approaches SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.8—testing for Max and Min, Optimisation. Points of Inflexion Questionbank

Unlock all 45 questions with a free account