Second Derivative Test

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

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

Example Determine areas of concavity for

Example Determine where the graph is concave up and concave down

Inflection Points Where a curve changes concavity

Example Determine Inflection Points for

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

Example Use the second derivative test to locate the extrema of