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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 profit from its signature pastry, which is modeled by the function $f(x) = x^3 - 6x^2 + 9x + 2$ , where $x$ represents the number of hours after opening and $f(x)$ represents the profit in hundreds of dollars.

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit 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 profit is at a local maximum or minimum.

[4]

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

[4]

Identify any local maximum or minimum points and justify your answer.

[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]

Determine the $x$ -coordinates of the local maximum and minimum points.

[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 $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[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 ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context 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 $f(x) = 3x^2 - 12x + 20$ , 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

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

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was 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

Consider the function $g(x) = \frac{2x - 5}{x + 1}$ defined for $x \in \mathbb{R}, x \neq -1$ .

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

[3]

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

[5]

Question 8

SLPaper 2

A biotechnology company is analyzing the population of a new strain of bacteria, which can be modeled by the function $g(x) = 5 e^{2x} - 4 e^x + 1$ , 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 point by solving the equation $g'(x) = 0$ .

[3]

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

[3]

Evaluate the function at the critical point to find the local minimum value of the bacterial population.

[2]

Determine the behavior of the bacterial population function as $x \to \infty$ and $x \to -\infty$ .

[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 2 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 ( $x \ge 0$ ).

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 potential maximum or minimum.

[3]

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

[4]

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

[3]

Paper

Level

Question 1

SLPaper 1

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit 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 profit is at a local maximum or minimum.

[4]

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

[4]

Identify any local maximum or minimum points and justify your answer.

[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]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

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

[3]

Question 3

SLPaper 2

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 $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[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 ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context of the company's profit.

[3]

Question 5

SLPaper 1

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

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

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was 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

Consider the function $g(x) = \frac{2x - 5}{x + 1}$ defined for $x \in \mathbb{R}, x \neq -1$ .

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

[3]

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

[5]

Question 8

SLPaper 2

A biotechnology company is analyzing the population of a new strain of bacteria, which can be modeled by the function $g(x) = 5 e^{2x} - 4 e^x + 1$ , 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 point by solving the equation $g'(x) = 0$ .

[3]

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

[3]

Evaluate the function at the critical point to find the local minimum value of the bacterial population.

[2]

Determine the behavior of the bacterial population function as $x \to \infty$ and $x \to -\infty$ .

[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 2 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]

[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 ( $x \ge 0$ ).

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 potential maximum or minimum.

[3]

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

[4]

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

[3]

Paper

Level

Question 1

SLPaper 1

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit 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 profit is at a local maximum or minimum.

[4]

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

[4]

Identify any local maximum or minimum points and justify your answer.

[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]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

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

[3]

Question 3

SLPaper 2

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 $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[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 ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context of the company's profit.

[3]

Question 5

SLPaper 1

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

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

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was 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

Consider the function $g(x) = \frac{2x - 5}{x + 1}$ defined for $x \in \mathbb{R}, x \neq -1$ .

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

[3]

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

[5]

Question 8

SLPaper 2

A biotechnology company is analyzing the population of a new strain of bacteria, which can be modeled by the function $g(x) = 5 e^{2x} - 4 e^x + 1$ , 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 point by solving the equation $g'(x) = 0$ .

[3]

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

[3]

Evaluate the function at the critical point to find the local minimum value of the bacterial population.

[2]

Determine the behavior of the bacterial population function as $x \to \infty$ and $x \to -\infty$ .

[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 2 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]

[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 ( $x \ge 0$ ).

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 potential maximum or minimum.

[3]

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

[4]

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

[3]

SL 5.2—Increasing and decreasing functions Questionbank