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Determining Intervals on Which a Function is Increasing or Decreasing

Hannah Hill

Hannah Hill

7 min read

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Study Guide Overview

This study guide covers how to determine the intervals where a function is increasing or decreasing using the derivative. It explains that a positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function. The guide emphasizes the importance of critical points (where the derivative is zero or undefined) in identifying these intervals. It provides a walkthrough example, step-by-step instructions, and practice problems to solidify understanding of this concept.

5.3 Determining Intervals on Which a Function is Increasing or Decreasing

What can the derivative of a function show us about the function itself? Can it tell us when the function increases or decreases? Yes, it can! In this lesson, we’ll delve into how we can use derivatives to determine when a function increases or decreases. 📈

🕑 When Does a Function Increase or Decrease?

In order to determine the intervals on which a function is increasing or decreasing, we first need to understand the concept of the derivative. The derivative of a function is the rate of change of the function at a given point.

Thus, we know the following:

  1. ➕=📈 If the derivative is positive at a certain point (which means the rate of change is positive at that point), the function is increasing at that point.
  2. ➖=📉 If the derivative is negative at a certain point, the function is decreasing at that point.

Take a look at this graph to see these trends in action. The gray line represents the function, ff, and the black line represents its derivative, f′f'.

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Graph of a function and its derivative to demonstrate the trend

Image Courtesy of Informal Calculus

Now, you may be thinking that finding the interval where a function is increasing or decreasing is as simple as finding the interval where the function’s derivative is positive or negative, respectively. Well, it is! 🪄

Where can a function change from increasing to decreasing and vice versa? It can only change its direction from increasing to decreasing and vice versa at its critical points, points where the function’s derivative equals 0 or is undefined, and the points where the function itself is undefined.

So, for each of the intervals defined by the points where the function can change behavior, we can determine whether the function ...