Increasing and Decreasing Intervals – Definition, Formulas

Have you wondered why the distance shortens as soon as you move towards your friend’s home? And why does it happen the other way round when you travel in the opposite direction? That is because of the functions. In calculus, increasing and decreasing functions are the functions for which the value of f (x) increases and decreases, respectively, with the increase in the value of x. 

To check the change in functions, you need to find the derivatives of such functions. If the value of the function increases with the value of x, then the function is positive. If the value of the function decreases with the increase in the value of x, then the function is said to be negative. 

Intervals of increase and decrease

Increasing and decreasing intervals of real numbers are the real-valued functions that tend to increase and decrease with the change in the value of the dependent variable of the function. To find intervals of increase and decrease, you need to determine the first derivative of the function. This is done to find the sign of the function, whether negative or positive. The function interval is said to be positive if the value of the function f (x) increases with an increase in the value of x. In contrast, the function interval is said to be negative if the value of the function f (x) decreases with the increase in the value of x. 

Alternatively, the interval of the function is positive if the sign of the first derivative is positive. The interval of the function is negative if the sign of the first derivative is negative. Hence, the positive interval increases, whereas the negative interval is said to be a decreasing interval. 

How do you write intervals of increase and decrease? 

You can represent intervals of increase and decrease by understanding simple mathematical notions given below: 

• The value of the interval is said to be increasing for every x < y where f (x) ≤ f (y) for a real-valued function f (x). 
• If the value of the interval is f (x) ≥ f (y) for every x < y, then the interval is said to be decreasing. 

You can also use the first derivative to find intervals of increase and decrease and accordingly write them.

• If the function’s first derivative is f’ (x) ≥ 0, the interval increases. 
• On the other hand, if the value of the derivative f’ (x) ≤ 0, then the interval is said to be a decreasing interval. 

Determining intervals of increase and decrease

Since you know how to write intervals of increase and decrease, it’s time to learn how to find intervals of increase and decrease. Let us learn how to find intervals of increase and decrease by an example. 

Consider a function f (x) = x³ + 3x² – 45x + 9. To find intervals of increase and decrease, you need to differentiate them concerning x. After differentiating, you will get the first derivative as f’ (x). 

Therefore, f’ (x) = 3x² + 6x – 45

Taking out 3 commons from the entire term, we get 3 (x²+ 2x -15). Now, finding factors of this equation, we get, 3 (x + 5) (x – 3). If you substitute these values equivalent to zero, you will get the values of x. 

Therefore, the value of x = -5, 3. 

To find the value of the function, put these values in the original function, and you will get the values as shown in the table below.

Interval	Value of x	f'(x)	Increasing/Decreasing
(-∞, -5)	x = -6	f'(-6) = 27 > 0	Increasing
(-5, 3)	x = 0	f'(0) = -45 < 0	Decreasing
(3, ∞)	x = 4	f'(4) = 27 > 0	Increasing

Therefore, for the given function f (x) = x³ + 3x² – 45x + 9, the increasing intervals are (-∞, -5) and (3, ∞) and the decreasing intervals are (-5, 3). 

Special Case: One-to-One function 

The strictly increasing or decreasing functions possess a special property called injective or one-to-one functions. This means you will never get the same function value twice. 

For example, you can get the function value twice in the first graph. However, in the second graph, you will never have the same function value. Hence, the graph on the right is known as a one-to-one function. 

This is useful because injective functions can be reversed. You can go back from a ‘y’ value of the function to the ‘x’ value. This is usually not possible as there is more than one possible value of x. 

Example 1: What will be the increasing and decreasing intervals of the function f (x) = -x³ + 3x² + 9? 

Solution: To find intervals of increase and decrease, you need to differentiate the function concerning x. Therefore, f’ (x) = -3x²  + 6x.

Now, taking out 3 common from the equation, we get, -3x (x – 2). To find the values of x, equate this equation to zero, we get, f'(x) = 0

⇒ -3x (x – 2) = 0

⇒ x = 0, or x = 2.

Therefore, the intervals for the function f (x) are (-∞, 0), (0, 2), and (2, ∞). To find the values of the function, check out the table below.

Interval	Value of x	f'(x)	Increasing/Decreasing
(-∞, 0)	x = -1	f'(-1) = -9 < 0	Decreasing
(0, 2)	x = 1	f'(1) = 3 > 0	Increasing
(2, ∞)	x = 4	f'(4) = -24 < 0	Decreasing

Hence, (-∞, 0) and (2, ∞) are decreasing intervals, and (0, 2) are increasing intervals.

Example 2: Do you think the interval (-∞, ∞) is a strictly increasing interval for f(x) = 3x + 5? If yes, prove that. 

Solution: To prove the statement, consider two real numbers x and y in the interval (-∞, ∞), such that x < y. 

Then, 3x < 3y. 

⇒ 3x + 5 < 3y + 5

⇒ f (x) < f (y)

Since, x and y are arbitrary values, therefore, f (x) < f (y) whenever x < y. Therefore, the interval (-∞, ∞) is a strictly increasing interval for f(x) = 3x + 5. Hence, the statement is proved. 

Example 3: Find whether the function f (x) x³−4x, for x in the interval [−1, 2] is increasing or decreasing. 

Solution: You need to start from -1 to plot the function in the graph. -1 is chosen because the interval [−1, 2] starts from that value. At x = -1, the function is decreasing. Once it reaches a value of 1.2, the function will increase. After the function has reached a value over 2, the value will continue increasing. With the exact analysis, you cannot find whether the interval is increasing or decreasing. So, let’s say within the interval [−1, 2],

• The curve decreases in the interval [−1, approx 1.2]
• The curve increases in the interval [approx 1.2, 2]

Determining intervals of increase and decrease using graph 

In the above sections, you have learned how to write intervals of increase and decrease. In this section, you will learn how to find intervals of increase and decrease using graphs. It would help if you examined the table below to understand the concept clearly. 

Increasing interval	Decreasing interval
The graph below shows an increasing function. This can be determined by looking at the graph given. Since the graph goes upwards as you move from left to right along the x-axis, the graph is said to increase.	The graph below shows a decreasing function. This can be determined by looking at the graph given. Since the graph goes downwards as you move from left to right along the x-axis, the graph is said to decrease.

Points to Ponder

• The function will yield a constant value and will be termed constant if f’ (x) = 0 through that interval.
• For a real-valued function f (x), the interval ‘I’ is said to be a strictly increasing interval if for every x < y, we have f (x) < f (y).
• For a real-valued function f (x), the interval ‘I’ is said to be a strictly decreasing interval if for every x < y, we have f (x) > f (y).
• For a function f (x), when x1 < x2 then f (x1) ≤ f (x2), the interval is said to be increasing. 
• For a function f (x), when x1 < x2 then f (x1) < f (x2), the interval is said to be strictly increasing. You have to be careful by looking at the signs for increasing and strictly increasing functions. 
• For a function f (x), when x1 < x2 then f (x1) ≥ f (x2), the interval is said to be decreasing. 
• For a function f (x), when x1 < x2 then f (x1) > f (x2), the interval is said to be strictly decreasing. 
• If the value of the function does not change with a change in the value of x, the function is said to be a constant function.