Section 4.4Concavity and the Second Derivative Test

Concavity

Concavity (cont.)

State the signs of and on the interval (0,2).Example 1

State the signs of and on the interval (0,2).Example 2

Determine the open intervals on which the graph is concave upward or concave downward.

Example 3

Determine the open intervals on which the graph is concave upward or concave downward.

Example 4

Determine the open intervals on which the graph is concave upward or concave downward.

Example 5

Examples:

Inflection Points

So . . . ◦Take the second derivative.◦Find out where it is zero or undefined.◦These are your possible inflection points.◦Only an inflection point if the concavity changes.

How do we find them?

Find the points of inflection and discuss the concavity of the graph of the function.

Example 6

Find the points of inflection and discuss the concavity of the graph of the function.

Example 7

Find the points of inflection and discuss the concavity of the graph of the function.

Example 8

The Second Derivative Test

Find all relative extrema. Use the Second Derivative Test where applicable.

Example 9

Find all relative extrema. Use the Second Derivative Test where applicable.

Example 10

Find all relative extrema. Use the Second Derivative Test where applicable.

Example 11

Be sure to be practicing the given problem sets!

Questions?