1. Second Derivative Test

2. Concavity
If the graph of a function f lies above all of its tangents, then it is called concave upward If the graph of a function f lies below all of its tangents, then it is called concave downward

3. Test for Concavity
If f’’ > 0 for all x in an interval, then the graph is concave upward If f’’ < 0 for all x in an interval, then the graph is concave downward

4. Example
Determine areas of concavity for

5. Example
Determine where the graph is concave up and concave down

6. Inflection Points
Where a curve changes concavity

7. Example
Determine Inflection Points for

8. Second Derivative Test
The second derivative can be used to identify local max and min values as well If f’ = 0 and the second derivative exists on an interval 1) If f’’(c) > 0, then this c is a minimum 2) If f’’(c) < 0, then this c is a maximum

9. Example
Use the second derivative test to locate the extrema of