Derivative sign and increasing/decreasing intervals | Year 11

01 / LEARN

The idea, step by step

Slope signs

A positive derivative means a graph rises as x increases; a negative derivative means it falls. This is about slope, not whether the graph itself is above or below an axis.

Find where the derivative is zero or undefined and split the domain at those inputs. For a continuous derivative, test one point in each interval to determine its sign there.

f^{'} (x) > 0 : increasing; f^{'} (x) < 0 : decreasing .

Sign charts

A factored derivative makes sign analysis efficient. Put the critical inputs in order, then record the signs between them. State open intervals for strict positive or negative derivative.

An isolated zero need not interrupt an overall increase. The graph of x³ increases through its horizontal tangent at 0. Do not assume that every derivative zero forces a turn.

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

A common trap

Thinking positive function values imply an increasing graph.

Instead: Increasing is determined by f′, not by the sign of f.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

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

$f^{'} (x) = 2 x - 4$ is 0 at 2.
The derivative is negative before 2 and positive after 2.

Decreasing on

(- \infty, 2)

; increasing on

(2, \infty)

Example 2

Worked example 2

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

Factor $g^{'} (x) = 3 (x - 1) (x + 1)$ .
Test −2, 0 and 2 to obtain +, − and +.

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

Check your understanding

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

Show answer

$x < 5 / 2$

Solve $5 - 2 x > 0$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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≠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 ↗

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