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SL 5.2—Increasing and decreasing functions - IB Questionbank | RevisionDojo

Practice SL 5.2—Increasing and decreasing functions with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A local bakery is analyzing the production of its signature pastry, which is modeled by the function $f(x) = x^3 - 6x^2 + 9x + 2$ , where $x$ represents the number of hours spent baking.

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the pastry production with respect to time.

[3]

Find the critical points of the function by solving $f^{\prime}(x) = 0$ . These points will help the bakery understand when the production rate is at a maximum or minimum.

[4]

Determine whether the function is increasing or decreasing on the intervals defined by the critical points. This will inform the bakery about the efficiency of production over time.

[4]

Identify any local maximum or minimum points and justify your answer. This analysis will help the bakery optimize its production schedule.

[4]

Question 2

SLPaper 2

A technology company is analyzing the performance of its new software product. The function representing the profit from the software is given by:

$f(x)=2 x^3-9 x^2+12 x+1$

Find the derivative $f^{\prime}(x)$ .

[2]

Solve for the critical points where $f^{\prime}(x)=0$ .

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Identify the local maximum and minimum values.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 3

SLPaper 2

A student is studying the decay of a chemical reactant over time in a laboratory experiment. The concentration of the reactant can be modeled by the exponential decay equation: $C(t) = C_0 e^{-k t}$ where $C(t)$ is the concentration at time $t$ , $C_0$ is the initial concentration, and $k$ is a positive constant representing the decay rate.

Show that the rate of change of concentration, $\frac{d C}{d t}$ , is always negative.

[3]

Evaluate the limit of $C(t)$ as $t \rightarrow \infty$ and interpret the result.

[3]

Sketch the graph of $C(t)$ against $t$ . Label the key features, including the initial concentration $C_0$ .

[4]

Find the half-life of the concentration, $t_{\frac{1}{2}}$ , in terms of $k$ .

[3]

Rewrite the equation in the form $\ln C(t) = \ln C_0 - k t$ and identify the gradient and intercept of the line.

[3]

Use the rewritten equation to determine the values of $C_0$ and $k$ if the data plotted shows a straight line with slope -0.7 and intercept 2.

[4]

Question 4

SLPaper 2

A startup company is analyzing its profit growth over time to make strategic decisions. The rate at which the company's profit is changing over time is given by:

$\frac{d P}{d t}=3 t^2-8 t$

where $P$ represents the company's profit (in thousands of dollars), and $t$ is the time in years since the company was founded.

Calculate the critical points for the profit rate and determine when the profit is increasing or decreasing.

[4]

Determine whether the profit is increasing or decreasing on the intervals defined by the critical points.

[4]

Interpret the meaning of the intervals of increase and decrease in terms of the company's profit.

[3]

Question 5

SLPaper 1

A local government is evaluating the cost function of a new public transportation initiative represented by the function $f(x) = 3x^2 - 12x + 5$ , where $x$ represents the number of buses in operation.

Find the derivative $f^{\prime}(x)$ .

[3]

Determine the critical points by setting $f^{\prime}(x) = 0$ .

[4]

Identify the intervals on which the function is increasing or decreasing.

[4]

Identify any local maxima or minima, if they exist.

[4]

Question 6

SLPaper 1

A company’s profit per year was found to be changing at a rate of

$\frac{d P}{d t} = 3 t^{2} - 8 t$

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the companywas founded, measured in years.

Determine whether the profit is increasing or decreasing when $t = 2$ .

[2]

One year after the company was founded, the profit was $4$ thousand dollars.

Find an expression for $P (t)$ , when $t \geq 0$ .

[4]

Question 7

SLPaper 1

A local café is studying the relationship between the price of a beverage and the number of customers it attracts, modeled by the function $g(x) = \frac{2x - 5}{x + 1}$ , where $x$ represents the price in dollars.

Calculate the derivative $g^{\prime}(x)$ to understand how the number of customers changes with price adjustments.

[3]

Determine the intervals on which the function is increasing or decreasing. This information will help the café make informed pricing decisions.

[5]

Question 8

SLPaper 2

A biotechnology company is analyzing the growth of a new strain of bacteria, which can be modeled by the function ${g(x) = 5 e^{2x} - 4 e^x}$ , where ${x}$ represents time in hours.

Calculate the derivative ${g'(x)}$ to determine the growth rate of the bacteria at any time ${x}$ .

[2]

Determine the critical points by solving the equation ${g'(x) = 0}$ .

[3]

Determine the intervals where the bacterial growth is increasing and decreasing.

[3]

Evaluate the function at the critical point to find the local extrema of the bacterial growth.

[2]

Explain the behavior of the bacterial growth function as ${x \rightarrow \infty}$ and ${x \rightarrow -\infty}$ .

[2]

Question 9

SLPaper 2

A function $f$ is given by $f(x) = (2x + 2)(5 - {x^2})$ .

The graph of the function $g(x) = {5^x} + 6x - 6$ intersects the graph of $f$ .

Expand the expression for $f(x)$ .

[1]

Find $f’(x)$ .

[3]

Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.

[3]

Find the exact value of each of the zeros of $f$ .

[3]

Write down the coordinates of the point of intersection.

[2]

Draw the graph of $f$ for $- 3 \leqslant x \leqslant 3$ and $- 40 \leqslant y \leqslant 20$ . Use a scale of 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 5 units on the $y$ -axis.

[4]

Question 10

SLPaper 2

A biotechnology company is analyzing the behavior of a new enzyme represented by the function $h(x) = e^{-x}(x^2 + 1)$ , where $x$ is the time in hours since the enzyme was introduced.

Use technology to find the derivative $h^{\prime}(x)$ to understand how the enzyme's activity changes over time.

[3]

Determine the critical points of the function to find when the enzyme's activity is at a maximum or minimum.

[4]

Identify the intervals where the enzyme's activity is increasing or decreasing.

[6]

State the local maximum or minimum values of the enzyme's activity, if any exist.

[4]

SL 5.2—Increasing and decreasing functions Questionbank