Increasing and Decreasing Intervals

The increasing or decreasing intervals of a function show where its graph rises or falls. We can find these intervals by checking the sign of the first derivative in each interval.

• If the first derivative is positive on an interval, the function is increasing there.

• If the first derivative is negative, the function is decreasing on that interval.
   

To write intervals of increase and decrease, you can follow these basic mathematical rules:

• An interval is said to be increasing if, for every x < y, the function satisfies f(x) ≤ f(y) for a real-valued function f(x).

• An interval is decreasing if, for every x < y, the function satisfies f(x) ≥ f(y).

You can also use the first derivative to identify these intervals:

• If the first derivative f'(x) ≥ 0 on an interval, the function is increasing there.

• If f'(x) ≤ 0, the function is decreasing on that interval.

This method makes it easier to analyze the behavior of functions and represent their increasing or decreasing nature.
 

Now that we understand what increasing and decreasing intervals mean, let’s learn how to find them for a function. We’ll use an example to understand the process step by step.

Consider the function:
f(x) = x³ + 3x² – 45x + 9

Step 1: Differentiate the function
Find the first derivative, f '(x):
f '(x) = 3x² + 6x – 45
= 3(x² + 2x – 15)
= 3(x + 5)(x – 3)

Step 2: Find critical points
Set f '(x) = 0:
3(x + 5)(x – 3) = 0
⇒ x = –5 and x = 3

Here, -5 and 3 are critical points. These are the points where the slope of the function changes.

Step 3: Identify intervals
Using the critical points, divide the number line into three intervals:
(–∞, –5), (–5, 3), and (3, ∞)
 

As a result, f(x) = x3 + 3x2 - 45x + 9 has increasing intervals (-∞, -5) and (3, ∞) and decreasing intervals (-5, 3).
 

A critical point of a function occurs when the first derivative of the function is either zero or undefined. These points are important for identifying extrema, which are the local maximum or minimum values of the function.

Critical Point: A point x = c is a critical point if either f '(c) = 0 or f '(c) is undefined.

Extrema: If the function changes from increasing to decreasing or vice versa around a critical point, then it is called an extrema — either a local maximum or a local minimum.

Let's examine a graph of a curve. The curve will turn at certain points, either from increasing to decreasing or the other way around. These points are called extrema. They may be:

• Local Maximum: The highest point in a small surrounding area.

• Local Minimum: The lowest point within a certain small area.
   

It becomes clear from the above figures that every extrema of a function is a point where its derivative changes sign. In other words, the function either goes from increasing to decreasing or vice versa. When identifying regions where a function is increasing or decreasing, it is important to examine the behavior around the extrema.

To find these intervals for any function f(x) over a given interval, follow these steps:

• Check if the function is differentiable and continuous in the given interval.

• Solve the equation f '(x) = 0. The solutions give the critical points (possible extrema).

• For each critical point x = c, examine the sign of the derivative f '(x) on either side of c to determine whether the function is increasing or decreasing in those intervals.
   

For a function to have increasing or decreasing intervals, it must be continuous and differentiable in that interval.

• Continuity means the graph of the function has no breaks or gaps at a point.

• Differentiability means the function’s first derivative exists at that point. 

Every differentiable function is continuous, but a continuous function may not be differentiable. 
 

In this section, we will learn how to visually represent increasing and decreasing intervals on a graph. This helps you identify these intervals simply by looking at the graph.

Below are two sample graphs of different functions. The first graph shows an increasing function because the curve goes up as we move from left to right along the x-axis. The second graph shows a decreasing function because the curve goes down as we move from left to right.
 

Increasing and decreasing intervals have real life applications in different fields. Here are few examples of applications - 

• Businesses utilize these intervals to monitor their earnings and determine when to increase or reduce output. For example, if profit rises with the number of items sold, the interval is increasing. If profit rises in proportion to the quantity sold, the function is increasing. If profit declines due to increased costs, the function is decreasing.

• In the health sector, this concept helps to track the effectiveness of a medicine over time. An interval that is increasing would indicate higher effectiveness, while one that is decreasing indicates less efficacy.

• Governments can use these intervals to examine population growth. This helps analyze whether population growth is increasing or decreasing. It enables them to plan for social welfare.