Your weak spots

Insights load after your first practice round.

Module 3 · L11 of 15 ~35 min +95 XP available

Increasing and Decreasing Functions

A function's direction tells you whether it is climbing or falling. The sign of the first derivative reveals this at a glance: positive means rising, negative means falling, zero means momentarily level. In this lesson you will master sign analysis, the core tool for sketching curves and solving optimisation problems.

Today's hook, You are driving up a mountain. The road climbs steeply, then flattens out, then plunges down the other side. If someone gave you the equation for the road's height, could you tell exactly where the climbing stops and the descent begins? The first derivative does this instantly.

0/5QUESTS

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall, your gut answer first

+5 XP warm-up

Before we start, what do you already know about when functions go up or down? How would you describe a function that is increasing at a particular point?

auto-saved

The two moves

+5 XP to read

There are only two moves in this entire lesson. Lock them into muscle memory and the rest is just calculation.

Move 1, Differentiate and find stationary points. Compute $f'(x)$, then solve $f'(x) = 0$ to identify the points that separate regions of different behaviour.

Move 2, Test signs in each interval. Pick a test value in each region between stationary points. If $f'(x) > 0$, the function is increasing there; if $f'(x) < 0$, it is decreasing.

$$f'(x) > 0 \Rightarrow \text{increasing} \qquad f'(x) < 0 \Rightarrow \text{decreasing}$$

Use open intervals

A function is neither increasing nor decreasing at a stationary point. Write $(2, \infty)$, not $[2, \infty)$.

Factorise first

Factorise $f'(x)$ fully before making a sign table. Unfactorised forms make sign analysis much harder.

Real-world link

A stock price “increasing” means its derivative is positive. The peak of the price graph is where the derivative is zero.

What you will master

Know

Key facts

$f'(x) > 0$ means increasing; $f'(x) < 0$ means decreasing
Stationary points are where $f'(x) = 0$
Intervals use open bracket notation at stationary points

Understand

Concepts

Why the sign of $f'(x)$ determines the direction of $f$
How stationary points separate intervals of increase from intervals of decrease
The connection between sign tables and curve sketching

Can do

Skills

Determine where a function is increasing or decreasing using the first derivative
Find intervals of increase and decrease algebraically
Use sign tables to analyse the behaviour of a function

Key terms

Increasing functionA function $f$ is increasing on an interval if $f'(x) > 0$ for all $x$ in that interval.

Decreasing functionA function $f$ is decreasing on an interval if $f'(x) < 0$ for all $x$ in that interval.

Stationary pointA point where $f'(x) = 0$, separating intervals of increase and decrease.

Sign tableA table listing test values and the sign of $f'(x)$ in each interval to determine increasing/decreasing behaviour.

Open intervalAn interval that does not include its endpoints, written $(a, b)$. Used for increasing/decreasing intervals.

First derivative$f'(x)$, the rate of change of $f$ at each point. Its sign tells us the direction of the function.

Increasing and decreasing functions

core concept

The sign of the first derivative tells us the direction of a function. When $f'(x) > 0$, the function is increasing (going uphill). When $f'(x) < 0$, it is decreasing (going downhill).

$$f'(x) > 0 \Rightarrow f \text{ is increasing} \qquad f'(x) < 0 \Rightarrow f \text{ is decreasing}$$

Stationary points occur where $f'(x) = 0$, and these points separate intervals of increase from intervals of decrease. A sign table for $f'(x)$ is the clearest way to show this analysis.

$f'(x) > 0$: increasing (teal). $f'(x) < 0$: decreasing (red). Stationary points at $f'(x) = 0$.

The sign table method

A sign table lists the critical $x$-values (where $f'(x) = 0$ or undefined) and the sign of $f'(x)$ in each interval between them:

Find $f'(x)$ and set it equal to zero to find stationary points.
Mark these points on a number line, dividing it into intervals.
Test one value of $x$ in each interval by substituting into $f'(x)$.
Record $+$ (increasing) or $-$ (decreasing) for each interval.

Open intervals matter. A function is neither increasing nor decreasing at a stationary point itself. Use open brackets: $(2, \infty)$ not $[2, \infty)$. In exams, this costs marks if you include the endpoint.

$f'(x) > 0$ on an interval $\Rightarrow$ $f$ is increasing on that interval; $f'(x) < 0$ on an interval $\Rightarrow$ $f$ is decreasing on that interval

Pause, copy the sign rules: $f'(x) > 0$ on an interval means $f$ is increasing there; $f'(x) < 0$ means $f$ is decreasing, into your book.

Quick check: True or false, a function can be classified as increasing at a stationary point where $f'(x) = 0$.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · QUADRATIC FUNCTION

Find where $f(x) = x^2 - 4x + 3$ is increasing and decreasing.

ApplyBand 3

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

Differentiate the function.

PROBLEM 2 · CUBIC FUNCTION

Find the intervals of increase and decrease for $f(x) = x^3 - 3x^2$.

ApplyBand 4

$f'(x) = 3x^2 - 6x = 3x(x - 2)$

Differentiate and factorise. Factorising is essential for accurate sign analysis.

PROBLEM 3 · REPEATED FACTOR

For $f(x) = x^4 - 4x^3$, find intervals of increase and decrease.

ApplyBand 4

$f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)$

Differentiate and factorise. Note the repeated factor $x^2$.

Quick check: For $f(x) = x^3 - 3x$, on which interval is $f$ decreasing?

Common errors · the 3 traps that cost marks

Trap 01

Including stationary points in the intervals

A function is neither increasing nor decreasing at a stationary point. Intervals of increase/decrease should use open intervals, e.g. $(2, \infty)$, not $[2, \infty)$.

Trap 02

Forgetting to factorise $f'(x)$ before testing signs

If $f'(x) = x^2 - 1$, factorise to $(x-1)(x+1)$ before making a sign table. Unfactorised forms make sign analysis much harder and lead to errors.

Trap 03

Confusing where $f'(x) = 0$ with where the function crosses the axis

$f'(x) = 0$ gives stationary points (turning points and inflections), not $x$-intercepts of $f$. The function may or may not cross the $x$-axis at a stationary point.

Odd one out: Three of the following are true statements about increasing/decreasing functions. Which one is NOT correct?

Work mode · how are you completing this lesson?

Quick-fire practice · 5 problems

Find where $f(x) = x^2 - 6x + 5$ is increasing.

Find the intervals where $f(x) = x^3 - 3x$ is decreasing.

For $f(x) = 2x^3 - 9x^2 + 12x$, find where the function is increasing.

Find where $f(x) = \frac{1}{x}$ is decreasing.

Sketch $y = x^3 - 3x^2$ showing intervals of increase and decrease.

Fill the blanks: drag each token into the matching blank.

positive negative zero open

When $f'(x)$ is ___, the function is increasing. When $f'(x)$ is ___, the function is decreasing. Stationary points occur where $f'(x)$ is ___, and we use ___ intervals to describe the regions of increase and decrease.

Match each function to its correct description of increasing/decreasing behaviour.

$f(x) = x^2$, interval $(-\infty, 0)$
$f(x) = x^2$, interval $(0, \infty)$
$f(x) = -x^2$, interval $(-\infty, 0)$
$f(x) = x^3$, interval $(-\infty, \infty)$

Always increasing ($f' = 3x^2 \ge 0$)
Increasing ($f' = -2x > 0$)
Increasing ($f' = 2x > 0$)
Decreasing ($f' = 2x < 0$)

Revisit your thinking

Earlier you were asked about when functions go up or down. The key insight: the first derivative is the key. Positive derivative means the function is increasing, negative means decreasing. Stationary points mark the exact boundaries between these regions, and we exclude them from the intervals.

auto-saved

Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidencethat tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 4

Q1. Find the intervals where $f(x) = x^3 - 6x^2 + 9x + 1$ is increasing. Show all working. 3 MARKS

auto-saved

ApplyBand 4

Q2. The derivative of a function is $f'(x) = (x - 1)(x + 2)^2$. Find where $f$ is increasing and where it is decreasing. Show all working. 3 MARKS

auto-saved

AnalyseBand 5

Q3. Find the values of $k$ for which $f(x) = x^3 + kx^2 + 3x$ is increasing for all $x$. 4 MARKS

auto-saved

📖 Comprehensive answers (click to reveal)

Drill 1: $f'(x) = 2x - 6 = 0 \Rightarrow x = 3$. Increasing on $(3, \infty)$.

Drill 2: $f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$. Decreasing on $(-1, 1)$.

Drill 3: $f'(x) = 6x^2 - 18x + 12 = 6(x-1)(x-2)$. Increasing on $(-\infty, 1)$ and $(2, \infty)$.

Drill 4: $f'(x) = -1/x^2 < 0$ for all $x \ne 0$. Decreasing on $(-\infty, 0)$ and $(0, \infty)$.

Q1 (3 marks): $f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3)$ [1]. $f'(x) = 0$ at $x = 1, 3$ [0.5]. Test signs: increasing on $(-\infty, 1) \cup (3, \infty)$ [1.5].

Q2 (3 marks): $(x+2)^2 \ge 0$ always, so sign depends on $(x-1)$ [0.5]. Decreasing for $x < 1$ (i.e. $f' < 0$ except at $x = -2$) [1]. Increasing for $x > 1$ [1]. Note: $x = -2$ is a stationary point of inflection [0.5].

Q3 (4 marks): $f'(x) = 3x^2 + 2kx + 3$ [0.5]. For always increasing, need $f'(x) > 0$ for all $x$, so discriminant $< 0$ [1]. $\Delta = 4k^2 - 36 < 0$ [1]. $k^2 < 9 \Rightarrow -3 < k < 3$ [1.5].

Boss battle · Sign Analyser

earn bronze · silver · gold

Five timed questions. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Arena coming soon

Science Jump · platform challenge

Climb platforms by answering increasing/decreasing questions. Lighter alternative to the boss.

Mark lesson as complete

Tick when you have finished the practice and review.

← Lesson 10 Lesson 12 · Areas →

Module overview · Maths Advanced · Checkpoint 1