5.3 Concavity and Second Derivatives

In this lesson, you will learn to classify a function's graph as concave up or concave down on an interval. You will also learn to find inflection points and local extrema using the Second Derivative Test.

Lesson Objectives

Upon completion of the lesson 5.3, you will be able to:

• • Find intervals where the graph of a function is concave up or concave down by applying the test for concavity that uses the sign of the second derivative to determine the intervals where the function is concave up or concave down.

  • Find inflection points.

  • Apply the Second Derivative Test to find local extrema.

Lesson Content

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

• 1. YouTube video: Concavity and Inflection Points

• 1. YouTube video: Second Derivative Test

• 1. Application of the Derivative: Lesson 29- Concavity & the Second Derivative Test [Internet Archive]

  2. The module includes concavity, the second derivative test, and homework.

1. Concavity and the Second Derivative Test [University of Houston]

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. Lecture 21: Second Derivative Test [Prairie State College] (using the second derivative to find intervals, finding the points of inflection, applying the second derivative test to classify relative extrema)

  2. Note: Not everyone uses the terminology "Hergert numbers" for the values of x where "(x) =0 or "(x) is undefined.

  3. Section 3.3: Second Derivative Tests [UC San Diego] (the second derivative test for concavity, inflection points, the second derivative test for local maxima and minima)

  4. Note: The existence of a tangent line is not always required in the definition of inflection point.

1. The Shape of a Graph - Part 2 [Paul's Online Math Notes]

   1. Concavity [McGraw Hill]

   2. Note: Skip example 5.6.

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. Practice Problems: Concavity and Points of Inflection [University of Houston] (answers and solutions provided)

  2. The Shape of a Graph - Part 2 [Paul's Online Math Notes]

  3. Note: Do problems #1 - 7, 9 - 13, 16.

  4. Drill - Graphs and Concavity [University of Tennessee]

  5. Note: Skip the 4th, 5th, and 9th problems.

Additional Resources

• 1. YouTube videos: Concavity and the 2nd Derivative Test

  2. Part 3.4a:

• 1. Part 3.4b:

1. Section 8.2 - Using the Second Derivative [Understanding Calculus]

   1. YouTube video: Finding Local Maximums/Minimums - Second Derivative Test

• 1. YouTube video: Concavity, Inflection Points, and Second Derivatives

• 1. Concavity and the Second Derivative Test [Harvey Mudd College] (some first derivative facts, concavity, concavity and the second derivative, inflection points, the second derivative test, key concepts: concavity theorem, the second derivative test)