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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

    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.

  3. 3

    Add one concrete example

    For f(x) = x^2 - 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).

  4. 4

    Avoid this incomplete answer

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

Definition

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.

Example

For f(x) = x^2 - 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).

Rule to remember

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.

Memory hook

Derivative positive means the graph climbs; derivative negative means the graph falls.

Examples and method

Worked example

Find intervals of increase and decrease for f(x) = x^3 - 6x^2 + 9x + 1. First f'(x) = 3x^2 - 12x + 9 = 3(x - 1)(x - 3). Critical points are x = 1 and x = 3. For x < 1, f'(x) > 0; for 1 < x < 3, f'(x) < 0; for x > 3, f'(x) > 0. Hence f increases on (-infinity, 1), decreases on (1, 3), and increases on (3, infinity).

Method to apply

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.

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

f'(x) = 3x^2 - 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).

Answer writing and exam use

1-mark answer

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-mark answer

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. 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. For f(x) = x^2 - 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-mark answer

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. 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. Find intervals of increase and decrease for f(x) = x^3 - 6x^2 + 9x + 1. First f'(x) = 3x^2 - 12x + 9 = 3(x - 1)(x - 3). Critical points are x = 1 and x = 3. For x < 1, f'(x) > 0; for 1 < x < 3, f'(x) < 0; for x > 3, f'(x) > 0. Hence f increases on (-infinity, 1), decreases on (1, 3), and increases on (3, infinity). Asked as interval-finding, sign-table reasoning, or a graph-based question where students identify where the curve rises or falls. Writing x = 1 and x = 3 as the intervals instead of using them to form intervals.
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