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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

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 5

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 6

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 7

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 8

SLPaper 2

Consider the function $f(t)=\frac{\sin (2 t)}{t}+\cos (t)$ , where $t \in \mathbb{R}, t \neq 0$ .

The graph of $y=f(t)$ has a local minimum at point $A$ in the interval $0<t<\pi$ . Find the coordinates of $A$ , giving the $t$ -coordinate to three decimal places.

[6]

Show that there is exactly one point of inflection, $B$ , on the graph of $y=f(t)$ for $t>0$ .

[6]

The coordinates of $B$ can be expressed in the form $B(a \pi, b)$ , where $a, b \in \mathbb{Q}$ . Find the values of $a$ and $b$ .

[3]

Question 9

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 10

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]

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