1. Concavity & the 2nd Derivative Test
Section 4

2. AP Exam Practice

3. Definition of Concavity
Let f be differentiable on an open interval, I. The graph of f is concave up on I if f’’ is positive and concave down on I if f’’ is negative.

4. Test for Concavity
Let f be a function whose 2nd derivative exists on an open interval, I.
If f’’(x) > 0, then f is concave up.
If f’’(x) < 0, then f is concave down.

5. Ex 1: Concavity
Determine the open intervals on which the graph of is concave up or down.

6. Ex 2: Concavity Determine the open intervals on which
is concave up or down.

7. HOMEWORK
Pg 189 #11-21 odds

8. Point of Inflection Point at which concavity changes. Theorem:
If (c, f(c)) is a point of inflection of the graph of f, then either f’’(c)=0 or f’’ does not exist at c.

9. Ex 1: Points of Inflection
Determine the points of inflection and intervals of concavity for
Determine the points of inflection and concavity for

10. Theorem: The 2nd Derivative Test
Let f be a function such that f’’(c)=0 and the 2nd derivative of f exists on an open interval containing c.
1. If f’’(c) > 0, then f(c) is a relative minimum.
2. If f’’(c) < 0, then f(c) is a relative maximum.
3. If f’’(c) = 0, then the test fails.

11. Ex 2: Relative Extrema
Find the relative extrema for

12. HOMEWORK
Pg 189 #11-21 odds (find pts. of inflection), 27 – 40 odds