How To Find Increasing And Decreasing Intervals On A

Amon Skres

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06-01-2025

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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 graph. This guide will walk you through the process, step-by-step.

Understanding Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points x₁ and x₂ within that interval, where x₁ < x₂, f(x₁) < f(x₂). Visually, this means the graph is sloping upwards from left to right.

Conversely, a function is decreasing on an interval if, for any two points x₁ and x₂ within that interval, where x₁ < x₂, f(x₁) > f(x₂). Graphically, this represents a downward slope from left to right.

The Crucial Role of the First Derivative

The key to finding these intervals lies in the first derivative of the function, denoted as f'(x) or df/dx. The first derivative represents the instantaneous rate of change of the function at any given point.

• If f'(x) > 0 on an interval, then f(x) is increasing on that interval. A positive derivative indicates a positive rate of change.
• If f'(x) < 0 on an interval, then f(x) is decreasing on that interval. A negative derivative signifies a negative rate of change.
• If f'(x) = 0, then f(x) has a critical point. This could be a local maximum, local minimum, or a saddle point. Further analysis (often using the second derivative test) is required to determine the nature of the critical point.

Step-by-Step Process

Let's outline the steps involved in finding increasing and decreasing intervals:

1. Find the first derivative: Calculate f'(x) using the appropriate differentiation rules (power rule, product rule, quotient rule, chain rule, etc.).

2. Find critical points: Set f'(x) = 0 and solve for x. These values of x are the critical points. Also, consider points where f'(x) is undefined (e.g., where the denominator of a fraction is zero).

3. Test intervals: The critical points divide the domain of the function into intervals. Select a test point within each interval and evaluate f'(x) at that point.

4. Determine increasing/decreasing intervals: Based on the sign of f'(x) in each interval, determine whether the function is increasing or decreasing on that interval.

Example

Let's consider the function f(x) = x³ - 3x + 2.

1. First derivative: f'(x) = 3x² - 3

2. Critical points: Set f'(x) = 0: 3x² - 3 = 0 => x² = 1 => x = ±1.

3. Test intervals: We have three intervals: (-∞, -1), (-1, 1), and (1, ∞).

   • Interval (-∞, -1): Test point x = -2. f'(-2) = 3(-2)² - 3 = 9 > 0. Therefore, f(x) is increasing on (-∞, -1).
   • Interval (-1, 1): Test point x = 0. f'(0) = 3(0)² - 3 = -3 < 0. Therefore, f(x) is decreasing on (-1, 1).
   • Interval (1, ∞): Test point x = 2. f'(2) = 3(2)² - 3 = 9 > 0. Therefore, f(x) is increasing on (1, ∞).

Conclusion: The function f(x) = x³ - 3x + 2 is increasing on the intervals (-∞, -1) and (1, ∞), and decreasing on the interval (-1, 1).

By following these steps, you can effectively determine the increasing and decreasing intervals of any differentiable function, providing valuable insight into its behavior and graphical representation. Remember to always carefully consider the domain of your function and any points where the derivative is undefined.