Increasing and decreasing functions AI HL

IB Mathematics: Applications and Interpretation HL · Unit 5: Calculus

Lesson 5.2 · Increasing and Decreasing Functions

Use the sign of the derivative to describe the behaviour of a function and identify turning points.

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Jesus calls us to carry the good news beyond our own circle. We can speak with humility and courage about his death for our sins, his resurrection and the life he offers. Our circumstances may rise and fall, but the message we are called to share remains good news for every person.

Learning objectives

By the end of this lesson, you should be able to:

identify intervals on which a function is increasing or decreasing from its graph;
use $f'(x)>0$ and $f'(x)<0$ to determine the behaviour of $f$ ;
read the sign of $f'(x)$ from a graph of the derivative;
find stationary points and classify them using a sign change;
recognise that $f'(x)=0$ does not always indicate a maximum or minimum;
interpret increasing and decreasing intervals in practical situations.

1. What do increasing and decreasing mean?

A function is increasing over an interval when its output rises as $x$ increases. It is decreasing when its output falls as $x$ increases.

Illustration of increasing and decreasing functions — Observe the direction of the graph as you move from left to right.

The derivative tells us the gradient of the curve at a particular input:

$f'(x)>0\ \Longrightarrow\ f\text{ is increasing locally}$

$f'(x)<0\ \Longrightarrow\ f\text{ is decreasing locally}$

Derivative	Tangent gradient	Behaviour of $f$
$f'(x)>0$	Positive	Graph rises from left to right
$f'(x)<0$	Negative	Graph falls from left to right
$f'(x)=0$	Zero	Horizontal tangent at that point

A derivative of zero describes the gradient at one point. To decide what happens on either side, examine the sign of the derivative in the surrounding intervals.

2. Reading the graph of $f$ and the graph of $f'$

When you are given the graph of $f$ , look for portions that rise or fall as you move from left to right. When you are given the graph of $f'$ , look at whether that graph lies above or below the $x$ -axis:

If the graph of $f'$ is above the $x$ -axis, $f'(x)>0$ and $f$ is increasing.
If the graph of $f'$ is below the $x$ -axis, $f'(x)<0$ and $f$ is decreasing.
Where the graph of $f'$ meets the $x$ -axis, $f'(x)=0$ . Check the signs on both sides before naming a turning point.

Worked example · Reading a derivative graph.

Suppose a graph of $f'$ shows that $f'(x)>0$ for $x<-2$ , $f'(x)<0$ for $-2<x<1$ , and $f'(x)>0$ for $x>1$ .

Then $f$ increases on $(-\infty,-2)$ , decreases on $(-2,1)$ , and increases on $(1,\infty)$ . The change from increasing to decreasing gives a local maximum at $x=-2$ . The change from decreasing to increasing gives a local minimum at $x=1$ .

The graph of $f'$ tells us the $x$ -coordinates of these points. Without additional information about $f$ , it does not tell us their $y$ -coordinates.

The first derivative sign test

Sign of $f'$ before	Sign of $f'$ after	Behaviour of $f$
Positive	Negative	Local maximum
Negative	Positive	Local minimum
Positive	Positive	No turning point from this sign test
Negative	Negative	No turning point from this sign test

A stationary point occurs where $f'(x)=0$ . It may be a local maximum, a local minimum or a stationary point where the graph continues in the same direction.

3. Finding intervals using a derivative

Consider the function:

$f(x)=x^3-4x$

Its derivative is:

$f'(x)=3x^2-4$

First find where the derivative is zero:

$3x^2-4=0 \quad\Longrightarrow\quad x=-\frac{2}{\sqrt3}\ \text{or}\ x=\frac{2}{\sqrt3}$

These values divide the real line into three intervals. Test one value in each interval.

Interval	Test value	Sign of $f'(x)$	Behaviour of $f$
$x<-2/\sqrt3$	$x=-2$	$f'(-2)=8>0$	Increasing
$-2/\sqrt3<x<2/\sqrt3$	$x=0$	$f'(0)=-4<0$	Decreasing
$x>2/\sqrt3$	$x=2$	$f'(2)=8>0$	Increasing

Therefore, $f$ is increasing on $(-\infty,-2/\sqrt3)$ and $(2/\sqrt3,\infty)$ , and decreasing on $(-2/\sqrt3,2/\sqrt3)$ .

The sign changes from positive to negative at $x=-2/\sqrt3$ , giving a local maximum. It changes from negative to positive at $x=2/\sqrt3$ , giving a local minimum.

$\text{Local maximum: } \left(-\frac{2}{\sqrt3},\frac{16}{3\sqrt3}\right)$

$\text{Local minimum: } \left(\frac{2}{\sqrt3},-\frac{16}{3\sqrt3}\right)$

4. A horizontal tangent does not always mean a turn

For $g(x)=x^3$ :

$g'(x)=3x^2$

At $x=0$ , $g'(0)=0$ . However, $3x^2$ is positive on both sides of zero. The function continues to increase through $(0,0)$ .

The point $(0,0)$ is a stationary point of inflection, not a local maximum or minimum. Always check for a change in sign instead of assuming that every solution of $f'(x)=0$ is a turning point.

5. Increasing and decreasing in context

Worked example · Visitors to an exhibition.

The number of visitors in a hall is modelled by $N(t)=120+30t-3t^2$ , where $t$ is the number of hours after opening and $0\leq t\leq8$ .

$N'(t)=30-6t$

For $0<t<5$ , $N'(t)>0$ , so the number of visitors is increasing. At $t=5$ , $N'(5)=0$ . For $5<t<8$ , $N'(t)<0$ , so the number is decreasing.

$N(5)=120+30(5)-3(5)^2=195$

The model predicts a maximum of $195$ visitors, five hours after opening. The derivative has units of visitors per hour; $N'(2)=18$ means the number present is increasing at a modelled rate of $18$ visitors per hour at that instant.

Using your TI-84 Plus CE

Enter $f(x)$ in Y= and choose a graph window covering the domain in the question.
Use 2nd → TRACE (CALC) → 4:maximum or 3:minimum to estimate a turning point. Set left and right bounds on either side of the point.
To check the gradient at a chosen input, use MATH → 8:nDeriv( or 2nd → TRACE (CALC) → 6:dy/dx.
Test gradients on both sides of a stationary point. A gradient of zero alone does not classify the point.
In an application, restrict your conclusions to the stated domain and include units.

IB-style Paper 1 · Skill practice

Show your reasoning. Mark allocations are shown in square brackets.

Question 1 · Reading derivative signs [5 marks]

A differentiable function $f$ has $f'(x)>0$ for $x<-2$ , $f'(x)<0$ for $-2<x<1$ , and $f'(x)>0$ for $x>1$ . Also, $f'(-2)=f'(1)=0$ .

State the intervals on which $f$ is increasing. [2]
State the interval on which $f$ is decreasing. [1]
Classify the stationary points at $x=-2$ and $x=1$ . [2]

Show solution

(a) $f$ is increasing on $(-\infty,-2)$ and $(1,\infty)$ . (b) It is decreasing on $(-2,1)$ . (c) The derivative changes from positive to negative at $x=-2$ , so there is a local maximum. It changes from negative to positive at $x=1$ , so there is a local minimum.

Question 2 · A polynomial function [8 marks]

Let $f(x)=x^3-4x$ .

Find $f'(x)$ . [2]
Find the values of $x$ for which $f'(x)=0$ . [2]
Determine the intervals on which $f$ is increasing and decreasing. [3]
State the $x$ -coordinate of the local maximum. [1]

Show solution

(a) $f'(x)=3x^2-4$ . (b) $x=\pm2/\sqrt3$ . (c) The derivative has signs $+$ , $-$ , $+$ across the intervals separated by these values. Hence $f$ increases on $(-\infty,-2/\sqrt3)$ and $(2/\sqrt3,\infty)$ , and decreases on $(-2/\sqrt3,2/\sqrt3)$ . (d) The local maximum occurs at $x=-2/\sqrt3$ .

Question 3 · A quadratic function [5 marks]

Let $h(x)=4x^2-8$ .

Find $h'(x)$ and the $x$ -coordinate of its stationary point. [2]
State where $h$ is increasing and decreasing. [2]
Give the coordinates of the turning point. [1]

Show solution

(a) $h'(x)=8x$ , so the stationary point is at $x=0$ . (b) Since $h'(x)<0$ for $x<0$ and $h'(x)>0$ for $x>0$ , the function decreases on $(-\infty,0)$ and increases on $(0,\infty)$ . (c) $h(0)=-8$ , so the minimum is $(0,-8)$ .

Question 4 · A stationary point without a turn [5 marks]

The derivative of a differentiable function is $p'(x)=(x-2)^2(x+1)$ .

Find the values of $x$ for which $p'(x)=0$ . [1]
Determine where $p$ is increasing and decreasing. [2]
Classify the stationary point at $x=-1$ and explain why $x=2$ is not a turning point. [2]

Show solution

(a) $x=-1$ and $x=2$ . (b) Since $(x-2)^2$ is positive except at $x=2$ , the sign of $p'$ away from its zeros is the sign of $x+1$ . Thus $p$ decreases on $(-\infty,-1)$ and increases on $(-1,2)$ and $(2,\infty)$ . It is also correct to describe $p$ as increasing throughout $(-1,\infty)$ . (c) The sign changes from negative to positive at $x=-1$ , giving a local minimum. The sign is positive on both sides of $x=2$ , so the graph does not turn there.

IB-style Paper 2 · Applications

Question 5 · Visitors to an exhibition [9 marks]

The number of visitors in an exhibition hall is modelled by $N(t)=120+30t-3t^2$ , where $t$ is measured in hours after opening and $0\leq t\leq8$ .

Find $N'(t)$ . [2]
Determine the intervals during which the number of visitors is increasing and decreasing. [3]
Find the maximum modelled number of visitors and when it occurs. [2]
Find $N'(2)$ and explain its meaning, including units. [2]

Show solution

(a) $N'(t)=30-6t$ . (b) The derivative is positive for $0<t<5$ and negative for $5<t<8$ . Thus the number increases during the first five hours and decreases thereafter. (c) $N(5)=195$ , so the maximum is $195$ visitors after $5$ hours. (d) $N'(2)=18$ visitors per hour: at that time, the number present is increasing at a modelled instantaneous rate of $18$ visitors per hour.

Question 6 · Profit from ticket sales [8 marks]

A theatre models its daily profit, in euros, by $P(q)=-0.04q^2+8q-160$ , where $q$ is the number of tickets sold and $0\leq q\leq180$ . Treat $q$ as continuous when using calculus.

Find $P'(q)$ . [2]
State the intervals over which the modelled profit increases and decreases. [2]
Find the number of tickets that maximises the modelled profit and calculate that profit. [2]
Calculate $P'(60)$ and interpret it in this context. [2]

Show solution

(a) $P'(q)=8-0.08q$ . (b) It is positive for $0<q<100$ and negative for $100<q<180$ . The profit increases and then decreases. (c) The maximum occurs at $q=100$ tickets, with $P(100)=-400+800-160=€240$ . (d) $P'(60)=€3.20$ per additional ticket. Near $60$ tickets, selling one more ticket increases the modelled profit by approximately €3.20.

Question 7 · Temperature over time [8 marks]

The temperature in a room is modelled by $T(t)=18+6t-t^2$ degrees Celsius, where $t$ is the number of hours after a heater is switched on and $0\leq t\leq6$ .

Find $T'(t)$ and state its units. [2]
Determine when the temperature is increasing and when it is decreasing. [2]
Find the maximum modelled temperature and the time at which it occurs. [2]
Find $T'(5)$ and explain the meaning of its negative sign. [2]

Show solution

(a) $T'(t)=6-2t$ , measured in degrees Celsius per hour. (b) The temperature increases for $0<t<3$ and decreases for $3<t<6$ . (c) $T(3)=18+18-9=27^\circ\mathrm C$ , the maximum, reached after $3$ hours. (d) $T'(5)=-4^\circ\mathrm C$ per hour: at $t=5$ , the modelled temperature is falling at an instantaneous rate of $4^\circ\mathrm C$ per hour.

Exam checklist

Find where $f'(x)=0$ , then test its sign in each interval. State increasing and decreasing intervals clearly. Use a change from positive to negative for a local maximum and from negative to positive for a local minimum. If the sign does not change, do not call the stationary point a turning point. In applications, check the stated domain and explain your result with units.

Lesson 5.2 · Increasing and Decreasing Functions

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Learning objectives

1. What do increasing and decreasing mean?

2. Reading the graph of $f$ and the graph of $f'$

The first derivative sign test

3. Finding intervals using a derivative

4. A horizontal tangent does not always mean a turn

5. Increasing and decreasing in context

Using your TI-84 Plus CE

IB-style Paper 1 · Skill practice

Question 1 · Reading derivative signs [5 marks]

Question 2 · A polynomial function [8 marks]

Question 3 · A quadratic function [5 marks]

Question 4 · A stationary point without a turn [5 marks]

IB-style Paper 2 · Applications

Question 5 · Visitors to an exhibition [9 marks]

Question 6 · Profit from ticket sales [8 marks]

Question 7 · Temperature over time [8 marks]

Exam checklist

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IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime

Curriculum

Increasing and decreasing functions AI HL

Tell others about Jesus

Learning objectives

1. What do increasing and decreasing mean?

2. Reading the graph of and the graph of

The first derivative sign test

3. Finding intervals using a derivative

4. A horizontal tangent does not always mean a turn

5. Increasing and decreasing in context

Using your TI-84 Plus CE

IB-style Paper 1 · Skill practice

Question 1 · Reading derivative signs [5 marks]

Question 2 · A polynomial function [8 marks]

Question 3 · A quadratic function [5 marks]

Question 4 · A stationary point without a turn [5 marks]

IB-style Paper 2 · Applications

Question 5 · Visitors to an exhibition [9 marks]

Question 6 · Profit from ticket sales [8 marks]

Question 7 · Temperature over time [8 marks]

Exam checklist

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2. Reading the graph of $f$ and the graph of $f'$