How to Find Intervals of a Continuously Decreasing Function

Increasing and Decreasing Intervals

Increasing and decreasing intervals are intervals of real numbers where the real-valued functions are increasing and decreasing respectively. To determine the increasing and decreasing intervals, we use the first-order derivative test to check the sign of the derivative in each interval. The interval is increasing if the value of the function f(x) increases with an increase in the value of x and it is decreasing if f(x) decreases with a decrease in x.

In this article, we will learn to determine the increasing and decreasing intervals using the first-order derivative test and the graph of the function with the help of examples for a better understanding of the concept.

1.	What are Increasing and Decreasing Intervals?
2.	Increasing and Decreasing Intervals Definition
3.	Finding Increasing and Decreasing Intervals
4.	Increasing and Decreasing Intervals Using Graph
5.	FAQs on Increasing and Decreasing Intervals

What are Increasing and Decreasing Intervals?

The intervals where the functions are increasing or decreasing are called the increasing and decreasing intervals. These intervals can be evaluated by checking the sign of the first derivative of the function in each interval. If the first derivative of a function is positive in an interval, then it is said to be an increasing interval and if the first derivative of the function is negative in an interval, then it is said to be a decreasing interval. Let us go through their formal definitions to understand their meaning:

Increasing and Decreasing Intervals Definition

The definitions for increasing and decreasing intervals are given below.

• For a real-valued function f(x), the interval I is said to be an 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 decreasing interval if for every x < y, we have f(x) ≥ f(y).

We can also define the increasing and decreasing intervals using the first derivative of the function f(x) as:

• If f'(x) ≥ 0 on I, then I is said to be an increasing interval.
• If f'(x) ≤ 0 on I, then I is said to be a decreasing interval.

Finding Increasing and Decreasing Intervals

Now, we have understood the meaning of increasing and decreasing intervals, let us now learn how to do calculate increasing and decreasing intervals of functions. We will solve an example to understand the concept better. Consider f(x) = x³ + 3x² - 45x + 9. Differentiate f(x) with respect to x to find f'(x).

f'(x) = 3x² + 6x - 45

= 3(x² + 2x - 15)

= 3 (x + 5) (x - 3)

Substitute f'(x) = 0

⇒ x = -5, x = 3

Now, the x-intercepts are of f'(x) are x = -5 and x = 3. The intervals that we have are (-∞, -5), (-5, 3), and (3, ∞). We will check the sign of f'(x) in each of these intervals to identify increasing and decreasing intervals.

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

Hence, the increasing intervals for f(x) = x³ + 3x² - 45x + 9 are (-∞, -5) and (3, ∞), and the decreasing interval of f(x) is (-5, 3).

Increasing and Decreasing Intervals Using Graph

We have learned to identify the increasing and decreasing intervals using the first derivative of the function. Now, we will determine the intervals just by seeing the graph. Given below are samples of two graphs of different functions. The first graph shows an increasing function as the graph goes upwards as we move from left to right along the x-axis. The second graph shows a decreasing function as the graph moves downwards as we move from left to right along the x-axis.

Important Notes on Increasing and Decreasing Intervals

• 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).
• The function is constant in an interval if f'(x) = 0 through that interval.

Related Topics

• Applications of Derivatives
• Differential Equations
• Calculus

Increasing and Decreasing Intervals Examples

1. Example 1: Determine the increasing and decreasing intervals for the function f(x) = -x³ + 3x² + 9.

   Solution: Differentiate f(x) = -x³ + 3x² + 9 w.r.t. x

   f'(x) = -3x² + 6x

   = -3x(x - 2)

   ⇒ f'(x) = 0

   ⇒ -3x(x - 2) = 0

   ⇒ x = 0, or x = 2

   The intervals that we have are (-∞, 0), (0, 2), and (2, ∞). We need to identify the increasing and decreasing intervals from these.

   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

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

2. Example 2: Show that (-∞, ∞) is a strictly increasing interval for f(x) = 3x + 5.

   Solution: Consider two real numbers x and y in (-∞, ∞) such that x < y. Then, we have

   x < y

   ⇒ 3x < 3y

   ⇒ 3x + 5 < 3y + 5

   ⇒ f(x) < f(y)

   Since x and y are arbitrary, therefore f(x) < f(y) whenever x < y.

   Answer: Hence, (-∞, ∞) is a strictly increasing interval for f(x) = 3x + 5.

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Increasing and Decreasing Intervals Questions

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FAQs on Increasing and Decreasing Intervals

What are Increasing and Decreasing Intervals in Algebra?

Increasing and decreasing intervals are intervals of real numbers where the real-valued functions are increasing and decreasing respectively.

Why are Only the X-values Used in Describing Increasing and Decreasing Intervals?

X-values are used to describe increasing and decreasing intervals because the values of the function f(x) increases or decreases with the increase in the x-values, i.e., the change in f(x) is dependent on the value of x.

How Do You Find Increasing and Decreasing Intervals of a Function?

We can find increasing and decreasing intervals of a function using its first derivative. We can find the critical points and hence, the intervals. Then, we can check the sign of the derivative in each interval to identify increasing and decreasing intervals.

How to Find Increasing and Decreasing Intervals Using Graph?

We can find increasing and decreasing intervals using a graph by seeing if the graph moves upwards or downwards as moves from left to right along the x-axis. For graphs moving upwards, the interval is increasing and if the graph is moving downwards, the interval is decreasing.

How Do you Know When a Function is Increasing?

A function f(x) is said to be increasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) ≤ f(y).

Which Function Does Not Have Increasing and Decreasing Intervals?

A constant function is neither increasing nor decreasing as the graph of a constant function is a straight line parallel to the x-axis and its derivative is always 0.

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Source: https://www.cuemath.com/algebra/increasing-and-decreasing-intervals/

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