How To Determine Increasing And Decreasing Intervals On A

Hannah Arendt

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31-12-2024

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Determining the intervals where a function is increasing or decreasing is a fundamental concept in calculus. Understanding this allows us to analyze the behavior of a function and its rate of change. This guide provides a clear and concise method for identifying these intervals.

Understanding Increasing and Decreasing Functions

A function is considered increasing on an interval if its value consistently rises as the input (x-value) increases. Conversely, a function is decreasing on an interval if its value consistently falls as the input increases. Visually, an increasing function climbs uphill from left to right, while a decreasing function descends downhill.

Methods for Identifying Increasing and Decreasing Intervals

There are two primary methods for determining these intervals: using a graph and using the first derivative.

1. Using the Graph

This is the most intuitive method. Simply examine the graph of the function:

• Increasing Intervals: Identify sections of the graph where the function's value is rising as you move from left to right.
• Decreasing Intervals: Identify sections where the function's value is falling as you move from left to right.

Important Note: Pay close attention to the x-values defining the boundaries of these increasing and decreasing sections. These x-values will define the intervals.

2. Using the First Derivative

The first derivative of a function, f'(x), provides crucial information about the function's increasing and decreasing behavior.

• f'(x) > 0: If the first derivative is positive at a point, the function is increasing at that point.
• f'(x) < 0: If the first derivative is negative at a point, the function is decreasing at that point.
• f'(x) = 0: If the first derivative is zero, the function has a critical point – this could be a local maximum, local minimum, or an inflection point.

Steps:

1. Find the first derivative: Calculate f'(x).
2. Find critical points: Solve for x when f'(x) = 0 or f'(x) is undefined. These points divide the x-axis into intervals.
3. Test intervals: Choose a test point within each interval and evaluate f'(x) at that point.
   • If f'(x) > 0, the function is increasing on that interval.
   • If f'(x) < 0, the function is decreasing on that interval.
4. State the intervals: Express the increasing and decreasing intervals using interval notation.

Example

Let's consider the function f(x) = x² - 4x + 3.

1. First derivative: f'(x) = 2x - 4

2. Critical points: 2x - 4 = 0 => x = 2

3. Test intervals:

   • Interval (-∞, 2): Test point x = 0, f'(0) = -4 < 0 (decreasing)
   • Interval (2, ∞): Test point x = 3, f'(3) = 2 > 0 (increasing)

4. Intervals:

   • Decreasing: (-∞, 2)
   • Increasing: (2, ∞)

By following these steps, you can accurately and efficiently determine the increasing and decreasing intervals of any given function, whether you are using a graphical approach or a calculus-based method. Remember to always clearly define the intervals using appropriate notation.