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 |
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:
The definitions for increasing and decreasing intervals are given below.
We can also define the increasing and decreasing intervals using the first derivative of the function f(x) as:
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) = x 3 + 3x 2 - 45x + 9. Differentiate f(x) with respect to x to find f'(x).
f'(x) = 3x 2 + 6x - 45
Substitute f'(x) = 0
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) = x 3 + 3x 2 - 45x + 9 are (-∞, -5) and (3, ∞), and the decreasing interval of f(x) is (-5, 3).
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
Related Topics
Example 1: Determine the increasing and decreasing intervals for the function f(x) = -x 3 + 3x 2 + 9. Solution: Differentiate f(x) = -x 3 + 3x 2 + 9 w.r.t. x f'(x) = -3x 2 + 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 |
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.
View Answer >
Math will no longer be a tough subject, especially when you understand the concepts through visualizations.
