Sec 3.4: Concavity and the Second Derivative Test

•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.

Definition of Concavity

Ex01: Determining Concavity

3

6)(

2 x

xf

Ex02: Determining Concavity

4

1)(

2

2

x

xxf

Points of Inflection

Ex03: Finding Points of Inflection

34 4)( xxxf

Determine the points of inflection and discuss the concavity of the graph of

Ex04: Using the Second Derivative Test

35 53)( xxxf

Find the relative extrema for

3.4 Problem Set A

• 1, 4, 7• 10, 13, 16, 19• 25, 27• 30, 33, 36, 39