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
1
You’re here
Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
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
2
You’re here
The two moves
Work through the core explanation before applying it.
02
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.
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.
3
You’re here
What you will master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
03
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
04
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.
4
You’re here
Increasing and decreasing functions
Work through the core explanation before applying it.
05
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.
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
5
You’re here
Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · QUADRATIC FUNCTION
Find where $f(x) = x^2 - 4x + 3$ is increasing and decreasing.
$f$ is increasing on $(-\infty, 0)$ and $(2, \infty)$; decreasing on $(0, 2)$.
PROBLEM 3 · REPEATED FACTOR
For $f(x) = x^4 - 4x^3$, find intervals of increase and decrease.
ApplyBand 4
1
$f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)$
Differentiate and factorise. Note the repeated factor $x^2$.
2
$f'(x) = 0$ when $x = 0$ or $x = 3$
Stationary points at $x = 0$ and $x = 3$.
3
$4x^2 \ge 0$ always (zero only at $x = 0$). Sign depends on $(x - 3)$. For $x < 3$: $(x - 3) < 0$ so $f'(x) \le 0$ → decreasing on $(-\infty, 3)$. For $x > 3$: $(x - 3) > 0$ so $f'(x) > 0$ → increasing on $(3, \infty)$.
Note: at $x = 0$, $f'(0) = 0$ but the function does not change direction. It is a stationary point of inflection, not a local extremum.
Quick check: For $f(x) = x^3 - 3x$, on which interval is $f$ decreasing?
Common errors · the 3 traps that cost marks
6
You’re here
Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
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?
Quick-fire practice · 5 problems
7
You’re here
Drill it, then lock it in
Run the quick drill and copy the summary into your book.
1
Find where $f(x) = x^2 - 6x + 5$ is increasing.
2
Find the intervals where $f(x) = x^3 - 3x$ is decreasing.
3
For $f(x) = 2x^3 - 9x^2 + 12x$, find where the function is increasing.
4
Find where $f(x) = \frac{1}{x}$ is decreasing.
5
Sketch $y = x^3 - 3x^2$ showing intervals of increase and decrease.
Fill the blanks: drag each token into the matching blank.
positivenegativezeroopen
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 ($f' = 2x < 0$)
12
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
1
You’re here
Multiple choice
Answer the drill bank and rate your confidence.
01
Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
2
You’re here
Short answer
Write full responses, then check them against the model answers.
02
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 $(3, \infty)$.
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].
1
You’re here
Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
01
Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
