5.2 Increasing and decreasing functions questions for IB Maths AA

36 questions on Increasing and decreasing functions, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 5 Calculus
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	22	1	23
Paper 2	12	1	13
Every paper	34	2	36

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00652Paper 1 · 15 marks

The gates of a festival ground open at 10:00. The number of people inside the ground, $N$ , $t$ hours after the gates open is modelled by

N (t) = 500 t^{2} (6 - t), 0 \leq t \leq 6.

(a)
Find $N^{'} (t)$ .
2 marks
(b)(i)
Find the value of $N^{'} (1)$ .
1 mark
(b)(ii)
Interpret your answer to part (b)(i) in this context.
2 marks
(c)
Find the greatest number of people inside the ground. Justify that your answer is a maximum.
4 marks
(d)(i)
Find the average rate of change of $N$ for $0 \leq t \leq 4$ .
2 marks
(d)(ii)
Find the exact values of $t$ at which the rate of change of $N$ is equal to the average rate of change found in part (d)(i).
4 marks

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Question 2 · AA-00340Paper 2 · 16 marks

The concentration of carbon dioxide in the air above a remote island rises steadily over the years, but also rises and falls with the seasons. The concentration, $C$ parts per million (ppm), is modelled by

C (t) = 412 + 2.1 t + 3.2 \sin (2 π t), 0 \leq t \leq 5,

where $t$ is the time in years since the start of 2020.

(a)(i)
Write down the concentration at the start of 2020.
1 mark
(a)(ii)
Find the concentration when $t = 1.25$ .
2 marks
(b)
Find $C^{'} (t)$ .
2 marks
(c)
Show that, according to the model, the concentration decreases during part of each year.
2 marks
(d)
Find the total length of time, for $0 \leq t \leq 5$ , during which the concentration is decreasing.
4 marks
(e)
Find the value of $t$ when the concentration first exceeds $421$ ppm.
3 marks
(f)
Show that $C (t + 1) - C (t)$ is constant, and interpret its value in context.
2 marks

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Question 3 · AA-00669Paper 1 · 5 marks

Consider the function $f (x) = 2 x^{3} - 9 x^{2} - 24 x + 5$ , where $x \in R$ .

(a)
Find $f^{'} (x)$ .
2 marks
(b)
Find the values of $x$ for which $f$ is decreasing.
3 marks

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Question 4 · AA-00454Paper 2 · HL only · 15 marks

A screening test is used for a disease, $D$ . The test gives a positive result for 94% of people who have the disease and for 3% of people who do not have the disease.

In a particular population, the proportion of people who have the disease is $p$ , where $0 < p < 1$ . A person is chosen at random from this population and tested.

(a)
Show that the probability that the person has the disease, given a positive result, is $\frac{0.94 p}{0.91 p + 0.03}$ .
3 marks
(b)
Find $f (0.01)$ .
1 mark
(c)
Show that $f$ is an increasing function.
3 marks
(d)
Find the value of $p$ for which $f (p) = 0.5$ .
2 marks
(e)
Find the set of values of $p$ for which a person with a positive result has a probability greater than 0.9 of having the disease.
3 marks
(f)
Find the probability that a person who has positive results in both tests has the disease.
3 marks

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Question 5 · AA-00671Paper 1 · 6 marks

Consider the function $f (x) = x^{3} - 3 x^{2} + 5 x - 2$ , where $x \in R$ .

(a)
Show that $f^{'} (x)$ can be written as $3 (x - 1)^{2} + 2$ .
3 marks
(b)
Hence explain why $f$ is increasing for all values of $x$ .
1 mark
(c)
Using your answer to part (b), explain why the equation $f (x) = 0$ has exactly one real solution, and show that this solution lies between $0$ and $1$ .
2 marks

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Question 6 · AA-00646Paper 2 · 16 marks

Two rooms are heated in different ways. The temperature, in °C, of room A, $t$ hours after the heating is switched on, is modelled by

T_{A} (t) = 25 - 13 e^{- 0.15 t}, 0 \leq t \leq 12.

(a)
Find the temperature of room A when the heating is switched on.
1 mark
(b)(i)
Find the value of $T_{A}^{'} (3)$ .
2 marks
(b)(ii)
State what your answer to part (b)(i) represents in this context.
1 mark
(c)(i)
Suppose that the model for room A remains valid for all $t \geq 0$ .
Write down $\lim_{t \to \infty} T_{A} (t)$ .
1 mark
(c)(ii)
Write down $\lim_{t \to \infty} T_{A}^{'} (t)$ .
1 mark
(d)
The temperature, in °C, of room B is modelled by
$T_{B} (t) = 12 + 0.8 t + 0.6 \sin (1.5 t), 0 \leq t \leq 12,$
where $1.5 t$ is in radians.
Find an expression for $T_{B}^{'} (t)$ .
2 marks
(e)
Find the total time, for $0 \leq t \leq 12$ , during which room B is warming at a greater rate than room A.
5 marks
(f)
Find the greatest difference between the temperatures of the two rooms for $0 \leq t \leq 12$ .
3 marks

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