Math 121

Section 3.3

Concavity

Concave Up

If a function is concave up, then is increasing on

Concave Down

If a function is concave down, then is decreasing on

Graph that is concave up on and concave down on

Test for Concavity

Let be a function whose second derivative exist on

1) If for all x on, then is concave up.

2) If for all x on, then is concave down.

3) If for all x on, then the test fails

Examples

1) Discuss the concavity of the function

Test for Concavity

Interval
Test Value	x = -1	x = 1
Sign of	Negative	Positive
Conclusion	Concave Down	Concave Up

2) Given the following function find all extrema points and test for concavity

Test for Extrema points

Interval
Test Value	x = -2	x = 0	x = 2
Sign of	Positive	Negative	Positive
Conclusion	Increasing	Decreasing	Increasing

Y coordinates of the critical points

Thus, the function will have a relative maximum at (-12 and a relative minimum at

(1,-2)

Concavity Test

Test for Concavity

Interval
Test Value	x = -1	x = 1
Sign of	Negative	Positive
Conclusion	Concave Down	Concave Up

3) Given the following function find all extrema points and test for concavity

Test for Extrema Points

Interval
Test Value	x = -1	x = 1
Sign of	Negative	Postive
Conclusion	Decreasing	Increasing

Y coordinate of critical point

Therefore, f has an absolute minimum at (0,2)

Concavity Test

4) Find all critical points and describe the concavity

Y coordinate of critical point

Test for Extrema Points

Interval
Test Value	x = -1	x = 1
Sign of	Negative	Postive
Conclusion	Decreasing	Increasing

f has a relative min at (0,0)

Concavity Test

Interval
Test Value	x = -1	x = 1
Sign of	Positive	Positive
Conclusion	Concave Up	Concave Up