1. 3.4: Concavity and the Second Derivative Test

2. Objectives
Determine intervals on which a function is concave upward or concave downward.
Find any points of inflection of the graph of a function.

3. Concavity Point of Inflection (point where the concavity changes)
Concave Up
Concave Down

4. Second Derivative f ’’(x)=f’(f’(x)) Slope of f’
f ’’ is positive if f ’ is increasing
(the slope of f is increasing)

5. Second Derivative f ’’(x)=f’(f’(x)) Slope of f ’
f ’’ is negative if f ’ is decreasing
(the slope of f is decreasing)

6. Theorem 3.7
If f ''(x)>0 for all x in I then the graph of f is concave upward in I.
If f ''(x)<0 for all x in I then the graph of f is concave downward in I.

10. Homework
3.4 (page 189): #1
5-11 odd
15, 17, 21