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, \(f'(x)\):

\(f '(x) = 3x² + 6x – 45\\ f '(x)= 3(x² + 2x – 15)\\ f '(x)= 3(x + 5)(x – 3)\)

Step 2: Find critical points

Set \(f '(x) = 0\):

\(3(x + 5)(x – 3) = 0\\ x+5 = 0\\ x = -5\\and, \\ x-3 = 0\\ x= 3\\ ⇒ x = –5 \ and\ x = 3\)

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, \(f(x) = x^3 + 3x^2 - 45x + 9\) 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, which may be:
 

• Local maximum: The highest point in a nearby region of the graph.
   
• Local minimum: The lowest point in a nearby region of the graph.

It becomes clear from the above figures that every extremum 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 \(f '(x) = 0\) to find the critical points, which are 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. 

While every differentiable function is continuous, not all continuous functions are 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.

Here are some of the tips and tricks for students to master increasing and decreasing intervals:
 

1. Understand the core idea, that it is an increasing Interval if the function goes up as 𝑥 increases. Decreasing Interval if the function goes down as 𝑥 increases.
    

2. Test intervals with simple numbers. Pick a number in each interval instead of complicated calculations. Only the sign of 𝑓′(𝑥) matters, not the exact value.
    

3. Use graphing for visualization. Draw a rough sketch of increasing sections slope upward, decreasing slope downward. For cubic or quartic functions, mark critical points and check slope signs. Visualization helps remember the logic rather than just formulas.
    

4. Remember that critical points are the key.
   
   Critical points = where derivative = 0 or undefined.
   
   These points divide the domain into intervals. Each interval has a consistent increasing or decreasing trend. Label intervals left to right, then test one number in each.
    

5. Try to connect the concepts with some real-life examples. For example, the speed of a car going uphill, where speed rises as time passes, is an example of increasing interval. At the same time, Battery percentage draining where battery drops as time passes, is an example of decreasing interval.

Increasing and decreasing intervals have real-life applications across various fields. Here are a few examples:
 

1. 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 with quantity sold, the interval is increasing; if it declines due to increased costs, the interval is decreasing.
    
2. 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.
    
3. 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.
    
4. Bank account balance checking uses this property. When money is deposited regularly, the balance rises and when withdrawals or payments happen, the balance drops
    
5. Stock price chart shows uptrends and downtrends is a perfect real-life illustration of increasing/decreasing functions. We can see an increasing interval, when the price of a stock rises throughout the day, and we can see a decreasing interval, when the price falls.