medium difficulty9 min

Increasing and Decreasing Functions

A function f is increasing on an interval when larger x-values give larger function values, and decreasing when larger x-values give smaller function values.

In one line: Derivative positive means the graph climbs; derivative negative means the graph falls.

Practice This Concept

Understand it

The sign of f'(x) tells the behaviour of f(x) on an interval where the function is differentiable.
If f'(x) > 0 throughout an interval, f is increasing there.
If f'(x) < 0 throughout an interval, f is decreasing there.
Critical points split the number line into intervals for sign testing.

Example

For f(x) = x² - 4x + 3, f'(x) = 2x - 4. Since f'(x) < 0 for x < 2 and f'(x) > 0 for x > 2, the function decreases on (-infinity, 2) and increases on (2, infinity).

Formula and rule

If f is differentiable on an interval I, then f'(x) > 0 for all x in I implies f is increasing on I, and f'(x) < 0 for all x in I implies f is decreasing on I.

Critical points occur where f'(x) = 0 or f'(x) does not exist, provided f itself is defined.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find intervals of increase and decrease for f(x) = x³ - 6x² + 9x + 1.

1First f'(x) = 3x² - 12x + 9 = 3(x - 1)(x - 3).
2Critical points are x = 1 and x = 3.
3For x < 1, f'(x) > 0; for 1 < x < 3, f'(x) < 0; for x > 3, f'(x) > 0.
4Hence f increases on (-infinity, 1), decreases on (1, 3), and increases on (3, infinity).

Method you can reuse

Differentiate f(x); solve f'(x) = 0 and note undefined derivative points; divide the domain into intervals; test the sign of f'(x) on each interval; write increasing and decreasing intervals using interval notation.

Visual tip

A number-line sign chart is useful: mark critical points, test signs of f'(x), and label intervals as increasing or decreasing.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For f(x) = x³ - 3x, on which interval is f increasing?

Think first, then show answer

f'(x) = 3x² - 3 = 3(x - 1)(x + 1). It is positive for x < -1 and x > 1, so f is increasing on (-infinity, -1) and (1, infinity).

Common wrong answer

Writing x = 1 and x = 3 as the intervals instead of using them to form intervals.

Don't mix this up

Checking the sign of f(x) instead of the sign of f'(x).

Exam tip

Make a derivative sign table using critical points; do not decide monotonicity from one random value unless it represents the whole interval.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as interval-finding, sign-table reasoning, or a graph-based question where students identify where the curve rises or falls.

1
Enough for 1 mark
A function f is increasing on an interval when larger x-values give larger function values, and decreasing when larger x-values give smaller function values.
2
Add for 2 marks
Example: For f(x) = x² - 4x + 3, f'(x) = 2x - 4. Since f'(x) < 0 for x < 2 and f'(x) > 0 for x > 2, the function decreases on (-infinity, 2) and increases on (2, infinity).
3
Add for 3 marks
The sign of f'(x) tells the behaviour of f(x) on an interval where the function is differentiable.

MCQ Quiz

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