How To Determine Increasing And Decreasing Intervals On A

Robert Patinson

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27-12-2024

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Determining the intervals where a function is increasing or decreasing is a fundamental concept in calculus. Understanding this helps in analyzing the behavior of a function and its overall shape. This guide will walk you through the process, using both graphical and analytical methods.

Understanding Increasing and Decreasing Functions

A function is increasing on an interval if its values consistently rise as the input (x-value) increases within that interval. Conversely, a function is decreasing on an interval if its values consistently fall as the input increases. A function can be both increasing and decreasing over different intervals of its domain.

Graphical Method: Identifying Intervals Visually

The easiest way to determine increasing and decreasing intervals is by visually inspecting the graph of the function.

• Increasing Intervals: Look for sections of the graph where the function's curve is going uphill from left to right. Imagine walking along the curve; if you're walking uphill, it's increasing.

• Decreasing Intervals: Look for sections where the curve is going downhill from left to right. If you're walking downhill, it's decreasing.

Important Note: Pay close attention to the x-values where the increasing or decreasing behavior begins and ends. These points define the boundaries of your intervals. Use interval notation (e.g., (a, b) representing all x such that a < x < b) or inequality notation (e.g., a < x < b) to express your answer.

Analytical Method: Using the First Derivative

A more precise method utilizes the first derivative of the function. The first derivative, f'(x), represents the instantaneous rate of change of the function at any given point.

• f'(x) > 0: If the first derivative is positive on an interval, the function is increasing on that interval.

• f'(x) < 0: If the first derivative is negative on an interval, the function is decreasing on that interval.

• f'(x) = 0: If the first derivative is zero, the function may have a local maximum or minimum at that point (critical point). Further analysis, such as the second derivative test, may be required to determine the nature of the critical point.

Steps for the Analytical Method:

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

2. Find critical points: Solve the equation f'(x) = 0 to find the x-values where the derivative is zero (or undefined). These are potential points where the function changes from increasing to decreasing or vice versa.

3. Test intervals: Choose test points within each interval created by the critical points. Substitute these test points into the first derivative. If f'(x) is positive, the function is increasing; if f'(x) is negative, the function is decreasing.

4. Express the intervals: Based on your analysis, express the increasing and decreasing intervals using interval or inequality notation.

Example

Let's consider the function f(x) = x³ - 3x.

1. First derivative: f'(x) = 3x² - 3

2. Critical points: 3x² - 3 = 0 => x² = 1 => x = ±1

3. Test intervals:

   • Interval (-∞, -1): Test point x = -2. f'(-2) = 3(-2)² - 3 = 9 > 0. The function is increasing.

   • Interval (-1, 1): Test point x = 0. f'(0) = -3 < 0. The function is decreasing.

   • Interval (1, ∞): Test point x = 2. f'(2) = 9 > 0. The function is increasing.

4. Increasing intervals: (-∞, -1) and (1, ∞)

5. Decreasing interval: (-1, 1)

By combining graphical and analytical methods, you can confidently and accurately determine the increasing and decreasing intervals of a function. Remember to always check your work and ensure your results are consistent with the visual representation of the graph.