Chapter 3.4 - Concavity and the Second Derivative Test

Objectives:

• • Determine intervals on which a function is concave upward or concave downward.

  • Find any points of inflection of the graph of a function.

  • Apply the Second Derivative Test to find relative extrema of a function.

Concavity & Points of Inflection (for Day #25 lesson)

Introduction to the concepts of concave up/down and points of inflection. Using the second derivative of a function to analytically determine concavity intervals and inflection points.

Second Derivative Test (for Day #26 lesson)

Applying the Second Derivative Test to find extrema of a function. Note, this test is NOT for finding inflection points or concavity intervals. The exercise was intentionally selected to demonstrate one weakness of the Second Derivative Test, but in many cases this test is still the quickest/easiest way for finding extrema. Here are the Second Derivative Test exercises.

Comparison of First Derivative Test, Second Derivative Test, Inflections Points (image)

Flowchart comparing the two tests used for finding extrema vs. the process for finding inflection points. This is a still image, not a video.

Exercise for relating f, f', and f" (image)

Here's the classroom exercise in which we relate descriptions of a function, its first derivative, and its second derivative. Click here or the image below for the exercise. Click here for the solution.

Ch 3.4 Test topics and format

Multiple Choice (Calculator allowed/required)

• • 15 questions

Free Response (Calculator allowed/required)

• • 5 questions

  • Intervals of Increasing/Decreasing

  • First Derivative Test

  • Intervals of Concave Up/Down

  • Inflection Points

  • Second Derivative Test

  • Drawing f' graphs, given the graph for f. Here's some more practice, both exercises and solutions provided.