How to Find Increasing and Decreasing Intervals

Given a function, f(x), we can determine the intervals where it is increasing and decreasing by using differentiation and algebra. 

Step 1: Find the derivative, f'(x), of the function. 

Step 2: Find the zeros of f'(x). Remember, zeros are the values of x for which f'(x) = 0. Set f'(x) = 0 and solve for x. 

Step 3: Determine the intervals. The intervals are between the endpoints of the interval of f(x) and the zeros of f'(x). If the interval of f(x) is not given, assume f(x) is on the interval (-∞, ∞). 

Step 4: Determine whether the function is increasing or decreasing on each interval. Given the interval (a, c), choose a value b, a < b < c. Solve for f'(b). If f'(b) is positive, f(x) is increasing on (a, c). If f'(b) is negative, f(x) is decreasing on (a, c).

Example: If g(x) = (x – 5)², find the intervals where g(x) is increasing and decreasing. 

Solution:

Step 1: Find the derivative of the function. 

Using the chain rule, 

g'(x) = 2(5 – x)

Step 2: Find the zeros of the derivative function.  

In other words, find the values of for which g(x) equals zero. You can do this by setting g(x) = 0 and using algebra to solve for x. From the definitions above, we know the function is constant at points where the derivative is zero. 

g'(x) = 0 = 2(5 – x) 

0 = 5 – x

x = 5

Step 3: Use the zeros to determine intervals.

Since x = 5 is the only zero for g'(x), there are just 2 intervals: from negative infinity to 5, and from 5 to negative infinity. 

These can be denoted in inequality notation: 

-∞ < x < 5

5 < x < ∞

Or in interval notation: 

(-∞, 5), (5, ∞)

Remember, the endpoints are NOT inclusive because g(x) is neither increasing nor decreasing at the endpoints. 

Step 4: Determine if the function is increasing or decreasing in each interval.

For the first interval, ((-∞, 5), we’ll choose b = 0. -∞ < x < 5

g'(b) = g'(0) = 2(5-0) = 10 

10 > 0 POSITIVE

For the second interval, (5, ∞), we’ll choose b = 6. 5 < 6 < ∞

g'(b) = g'(6) = 2(5-6) = -2 

-2 < 0 NEGATIVE

Therefore, g(x) is increasing on (-∞, 5) and decreasing on (5, ∞). We can verify our results visually. In the graph below, you can clearly see that f(x) = (x – 5)² is increasing on the interval (5, ∞) and decreasing on the interval (-∞, 5).

We can visually verify our result by investigating the graph of g(x). 

Looking at the graph, g(x) is indeed increasing in the interval from negative infinity to 5 and decreasing in the interval from 5 to infinity. 

Example: Find the intervals in -20 < x < 20 where g(x) is increasing and decreasing given g'(x) = x² – 100.

Solution:

If the derivative is given, we can skip the first step and go straight to finding the zeroes. 

g'(x)= 0 = x² – 100

x² = 100

[Tex]x = 10, -10[/Tex]

Intervals: (-20, -10), (-10, 10), (10, 20) 

For (-20, -10), we’ll choose b = -12. -20 < -12 < -10 

g'(-12) = 44 > 0 

For (-10, 10), we’ll choose b = 0. -10 < 0 < 10 

g(0) = -100 < 0 

For (10, 20), we’ll choose b = 12. 10 < 12 < 20 

g(12) = 44 > 0

Hence, for -20 < x < 20, g(x) is increasing on (-20, -10) and (10, 20) and decreasing on (-10, 10). 

Read More,

• Rate Change of Quantities
• Approximation
• Tangents and Normals Lines

If you’re studying calculus, then you’re probably familiar with the concepts of increasing and decreasing functions. These terms refer to the behavior of a function as its input values change. Specifically, an increasing function is one that becomes larger as its input values increase, while a decreasing function is one that becomes smaller as its input values increase. Understanding these concepts is crucial for solving a variety of calculus problems, from finding maximum and minimum values to understanding the behavior of graphs. 

In this article, we’ll delve deeper into increasing and decreasing functions, exploring how to identify them, and how to use them to solve problems in calculus.