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SL 5.8—Testing for max and min, optimisation. Points of inflexion - IB Questionbank | RevisionDojo

Practice SL 5.8—Testing for max and min, optimisation. Points of inflexion 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

The equation of a curve is $y = \frac{1}{2}x^4 - \frac{3}{2}x^2 + 7$ .

The gradient of the tangent to the curve at a location Q is $-10$ .

Find ${\frac{{dy}}{{dx}}}$ .

[2]

Find the coordinates of Q.

[4]

Question 2

SLPaper 1

A sustainable fashion company produces and sells eco-friendly clothing. The profit, in dollars, from selling x units of a particular clothing item is given by:

${P(x) = 100x - 5x^2}$

Determine the number of units x that maximises the profit.

[4]

Calculate the maximum profit.

[3]

Determine whether the function P(x) has a point of inflection, showing all necessary steps.

[8]

Question 3

HLPaper 1

Consider a water tank in the shape of an inverted cone with a height of 10 meters and a base radius of 5 meters. Water is being pumped into the tank at a rate of 3 cubic meters per minute.

Find the rate at which the water level is rising when the water is 4 meters deep.

[5]

Question 4

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 5

SLPaper 1

A sculptor sells $x$ sculptures per month. Her monthly profit in Canadian dollars (CAD) can be modelled by $P\left(x\right) = -\frac{1}{5}x^3 + 7x^2 - 120,\,\,x \geqslant 0.$

Differentiate $P(x)$ .

[2]

Hence, find the number of sculptures that will maximize the profit.

[3]

Find the value of $P$ if no sculptures are sold.

[1]

Question 6

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 7

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 $V = \pi h (R^2 - h^2)$ .

[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 8

HLPaper 2

A rectangular piece of cardboard with dimensions $24$ cm by $36$ cm is to be made into an open box by cutting squares of side length x cm from each corner and folding up the sides.

Find the volume $V$ of the box in terms of $x$ .

[2]

Determine the value of $x$ that maximizes the volume of the box.

[3]

Verify that the value of x found in part 2 is a maximum by using the second derivative test.

[2]

Calculate the maximum volume of the box.

[2]

Question 9

SLPaper 2

A company is designing a new logo in the shape of a rectangle with a semicircle on top. The width of the rectangle is twice its height. The semicircle has a diameter equal to the width of the rectangle. The total area of the logo must be 200 cm².

Let the height of the rectangle be $h$ cm. Write an expression for the width of the rectangle in terms of $h$ .

[1]

Write an expression for the area of the rectangle in terms of $h$ .

[1]

Write an expression for the area of the semicircle in terms of $h$ .

[2]

Form an equation in $h$ and solve it to find the height of the rectangle. Give your answer to 3 significant figures.

[3]

Calculate the width of the rectangle. Give your answer to 3 significant figures.

[1]

Question 10

HLPaper 2

Consider the function $g(x) = \frac{\sqrt x }{\sin x}$ , $0 < x < \pi$ .

Consider the region bounded by the curve $y = g(x)$ , the $x$ -axis and the lines $x = \frac{\pi }{6}, x = \frac{\pi }{3}$ .

This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.

[3]

Sketch the graph of $y = g(x)$ showing clearly the minimum point and any asymptotic behaviour.

[3]

Determine the values of ${x}$ for which ${g(x)}$ is a decreasing function.

[2]

Show that the $x$ -coordinate of the minimum point on the curve $y = g(x)$ satisfies the equation $\tan(x) = 2x$ .

[5]

Find the coordinates of the point on the graph of $g$ where the normal to the graph is parallel to the line $y = -x$ .

[4]

SL 5.8—Testing for max and min, optimisation. Points of inflexion Questionbank