Increasing and Decreasing Functions — Applications of Derivatives

1. What It Means for a Function to Increase or Decrease

A function is said to be increasing when its output rises as x increases. It is decreasing when its output falls as x increases. In simple notes-style: increasing means the graph goes upward, decreasing means it goes downward.

2. Derivative as an Indicator

The derivative gives the slope of the tangent line. The sign of the derivative tells whether the graph is moving upward or downward at a point.

2.1. Conditions

If \( f'(x) > 0 \), the function is increasing.
If \( f'(x) < 0 \), the function is decreasing.
If \( f'(x) = 0 \), the point may be a peak, valley, or flat region.

3. Critical Points

A critical point occurs where the derivative becomes zero or undefined. These points split the number line into intervals where the function might change its behaviour from increasing to decreasing or vice versa.

4. Finding Intervals of Increase and Decrease

The typical process involves:

4.1. Step-by-Step

Find the derivative f'(x).
Solve f'(x) = 0 to identify critical points.
Check the sign of f'(x) in each interval between critical points.
Positive derivative → increasing; negative → decreasing.

5. Graphical Understanding

The slope of the tangent line tells how the graph behaves:

Slope positive → curve rising
Slope negative → curve falling
Slope zero → flat point

6. Examples

These examples show how the derivative helps in identifying behaviour.

6.1. Example 1 — Polynomial

f(x) = x^3 - 3x

Derivative:

f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1)

Critical points: x = -1, x = 1

Checking intervals:

x < -1 → f'(x) > 0 → increasing
-1 < x < 1 → f'(x) < 0 → decreasing
x > 1 → f'(x) > 0 → increasing

6.2. Example 2 — Square Root Function

f(x) = \sqrt{x}