Derivative sign and increasing/decreasing intervals | Year 11

01 / LEARN

Understand the idea.

Slope signs

Read the derivative as a slope, not a height. Where $f^{'} (x) > 0$ throughout an interval, the graph rises as input increases; where $f^{'} (x) < 0$ , it falls. A graph can be below the horizontal axis and still be increasing.

First locate derivative zeros and any inputs where the derivative is undefined. These divide the domain into candidate intervals. For polynomial derivatives, continuity and the absence of further zeros mean the sign cannot change inside any one of these intervals.

\begin{aligned} f^{'} (x) > 0 & : increasing \\ f^{'} (x) < 0 & : decreasing \end{aligned}

Sign charts

Factor the derivative when possible, order its roots, and determine the sign of each factor between adjacent roots. A test value can identify an interval’s sign once you know there are no missing zeros or discontinuities. State the intervals in increasing input order.

A derivative zero identifies a horizontal tangent, not necessarily a change in direction. For $f (x) = x^{3}$ , the derivative $3 x^{2}$ is positive on both sides of zero. The function continues increasing through that isolated stationary input.

Sign chart for f(x)=x³−3x
Interval	x<−1	−1<x<1	x>1
Sign of derivative	+	−	+
Behaviour	Increasing	Decreasing	Increasing

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Find the monotonic intervals of $f (x) = x^{2} - 4 x + 1$ .

Worked solution

1
Differentiate and locate the derivative zero.
$\begin{aligned} f^{'} (x) & = 2 x - 4 = 2 (x - 2) \\ f^{'} (x) = 0 & ⟹ x = 2 \end{aligned}$
This is the only input where the continuous linear derivative can change sign.
2
Determine the signs on both intervals.
$\begin{aligned} x < 2 & ⟹ 2 (x - 2) < 0 \\ x > 2 & ⟹ 2 (x - 2) > 0 \end{aligned}$
The positive factor 2 leaves the sign determined by whether the input is below or above 2.
3
Translate derivative signs into graph behaviour.
$\begin{aligned} decreasing: & (- \infty, 2) \\ increasing: & (2, \infty) \end{aligned}$
Negative slope means falling and positive slope means rising as the input increases.

Answer

Decreasing on $(- \infty, 2)$ ; increasing on $(2, \infty)$ .

EXAMPLE 02

Worked example 2

Analyse the sign of $g^{'} (x) = 3 x^{2} - 3$ .

Worked solution

1
Factor the derivative and order its roots.
$g^{'} (x) = 3 (x - 1) (x + 1), x = - 1, 1$
The two roots divide the real line into three intervals with no further derivative zeros.
2
Determine each factor’s sign.
$\begin{aligned} x < - 1 : & (-) (-) = + \\ - 1 < x < 1 : & (-) (+) = - \\ x > 1 : & (+) (+) = + \end{aligned}$
The factor 3 is positive, so the signs of the two linear factors determine the product on each whole interval.
3
State the monotonic behaviour.
$\begin{aligned} increasing: & (- \infty, - 1) and (1, \infty) \\ decreasing: & (- 1, 1) \end{aligned}$
The graph rises where its derivative is positive and falls where its derivative is negative.

Answer

Increasing for x<−1 and x>1; decreasing for −1<x<1.

03 / YOUR TURN

Try it yourself.

Check your understanding

If $f^{'} (x) = 5 - 2 x$ , where is f increasing?

Show answer & reasoning

$x < 5 / 2$

Increasing requires $5 - 2 x > 0$ . Rearranging gives $- 2 x > - 5$ , hence $x < \frac{5}{2}$ after reversing the inequality when dividing by $- 2$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Find the increasing/decreasing intervals of $f (x) = x^{2} + 6 x$ .
3 marks
Analyse $f^{'} (x) = 4 - 2 x$ .
3 marks
Find the sign intervals of $f^{'} (x) = 3 (x - 2) (x + 1)$ .
3 marks
If $g^{'} (x) = - (x - 1) (x - 4)$ , where is $g$ increasing?
3 marks
$h^{'} (x) = 3 x^{2}$ . Is $h$ decreasing anywhere?
3 marks
A graph is below the x-axis but has positive tangent gradients. Is it increasing or decreasing?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the increasing/decreasing intervals of $f (x) = x^{2} + 6 x$ .
3 marks
$f^{'} (x) = 2 x + 6$ changes from − to + at $- 3$ .
Decreasing $x < - 3$ ; increasing $x > - 3$ .
Analyse $f^{'} (x) = 4 - 2 x$ .
3 marks
The derivative is positive before 2 and negative after 2.
Increasing $x < 2$ ; decreasing $x > 2$ .
Find the sign intervals of $f^{'} (x) = 3 (x - 2) (x + 1)$ .
3 marks
Roots are $- 1$ and 2. Testing $- 2$ , 0 and 3 gives +, −, +.
Positive $x < - 1$ or $x > 2$ ; negative $- 1 < x < 2$ .
If $g^{'} (x) = - (x - 1) (x - 4)$ , where is $g$ increasing?
3 marks
The leading minus gives derivative signs −, +, − across 1 and 4.
$1 < x < 4$
$h^{'} (x) = 3 x^{2}$ . Is $h$ decreasing anywhere?
3 marks
$3 x^{2}$ is positive for $x \neq 0$ and 0 at $x = 0$ ; it is never negative.
No. It is increasing on both sides of 0.
A graph is below the x-axis but has positive tangent gradients. Is it increasing or decreasing?
3 marks
Its values rise as $x$ increases, even while those values remain negative.
Increasing

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topic 6: rates of change, derivatives and applications. Introductory polynomial calculus with explicit modelling and interpretation.

Official curriculum source ↗

In this lesson

Understand the idea.

Slope signs

Sign charts

See the method in action.

Worked example 1

Differentiate and locate the derivative zero.

Determine the signs on both intervals.

Translate derivative signs into graph behaviour.

Worked example 2

Factor the derivative and order its roots.

Determine each factor’s sign.

State the monotonic behaviour.

Try it yourself.

Check your understanding

Keep practising