5.1 Maxima and Minima

In this lesson, you will learn about extreme values of a function: whether they exist, how to classify them, and how to find them.

Lesson Objectives

Upon completion of the lesson 5.1, you will be able to:

• • Distinguish between maximum and minimum values of a function.

  • Distinguish between absolute (or global) and relative (or local) maximum or minimum values of a function.

  • Identify absolute and local maximum and minimum values of a function from a graph.

  • Recognize that a function may have no absolute or relative (or local) maximum or minimum value.

  • Find critical points (critical numbers) of a function.

  • Use analytical methods to locate the absolute maximum and minimum value of a continuous function on a closed interval.

  • Recognize that if a function has a local maximum or minimum value at x=a, then the derivative at x=a is either 0 or does not exist.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. Maximum and minimum values [University of New England]

  2. Note: Watch the video first. Then, try to solve the 10 problems listed below the video screen. When you click on "Answer" or "Explanation", an appropriate response will be shown so that you can check your answers and explanations. Skip problem #6.

  3. YouTube video: Maxima and Minima

• 1. YouTube video: Critical Points

• 1. YouTube video: Maximum and Minimum Values

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Critical Points [Paul's Online Math Notes] Skip examples 5, 6, and 7.

1. Maximum and minimum values [Paul's Online Math Notes]

   1. Chapter 3: Applications of differentiation - Maximum and minimum value [McGraw-Hill]

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

1. Maximum and minimum values [Paul's Online Math Notes]

   1. Critical Points [Paul's Online Math Notes]

   2. Note: Skip Problems #14 - 17.

   3. Drill - Maxima and Minima [University of Tennessee] (on a closed interval)

   4. Note: Skip the 6th and 8th problems that use exponentials or logarithms.

   5. Finding Absolute Extrema [Paul's Online Math Notes]

   6. Note: Skip Problems # 12 & 13.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. Relative Extrema and Critical Numbers [Internet Archive]

  2. Note: There are two lessons at this site. View only the first lesson titled "Relative Extrema and Critical Numbers."

1. Extreme Values of Functions

   1. YouTube video: Local and Absolute Maximum and Minimum from a Graph

• 1. YouTube video: Maxima and Minima