1. Second Derivative Test
4.3b Concavity And the
Second Derivative Test
State Standard
– 9.0a Students find the intervals in which the graph of functions is concave up and concave down.
– 9.0c Students can identify points of inflection of graphs of functions.
Objective – To be able to use the 2nd derivative test to find concavity and points of inflection.

2. In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.

3. First derivative:
is positive
Curve is rising.
is negative
Curve is falling.
is zero
Constant - Possible local maximum or minimum.
Second derivative:
is positive
Curve is concave up.
is negative
Curve is concave down.
is zero
Possible inflection point
(where concavity changes).

4. Possible inflection point at .
Example:
We will use the 2nd derivative test to find the Concavity of the function.
Possible inflection point at
negative
Curve is concave down.
positive
Curve is concave up.
inflection point at

5. Possible inflection point at .
Example:
We will use the 2nd derivative test to find the Concavity of the function.
Possible inflection point at
negative
Curve is concave down.
positive
inflection point at
Curve is concave up.

6. Possible inflection point at
Example:
We will use the 2nd derivative test to find the Concavity of the function.
Possible inflection point at
positive
Curve is concave up.
negative
Curve is concave down.
inflection point at
positive
Curve is concave up.

7. Find the intervals of concavity and the inflection point for
4.3b Assignment
Find the intervals of concavity
and the inflection point for
# 33 – 37 on pg. 305