1. 1 2 Sec 4.3: Concavity and the Second Derivative Test
the curve lies above the tangents
the curve lies below the tangents
Concavity Test
Concavity Test 1
f ’’(x) > 0 for all x in I 2
f ’’(x) < 0 for all x in I
f(x) concave Up
f(x) concave Down

3. 1 2 Sec 4.3: Concavity and the Second Derivative Test Concavity
f ’’(x) > 0 for all x in I 2
f(x) concave Up
Concavity
f ’’(x) < 0 for all x in I
f(x) concave Down

4. Sec 4.3: Concavity and the Second Derivative Test
Example:
Study the concavity of the function

5. Sec 4.3: Concavity and the Second Derivative Test

6. _ _ + + 1 2 Sec 4.3: Concavity and the Second Derivative Test
Inflection point: 1 _ + _ 2 +

7. Sec 4.3: Concavity and the Second Derivative Test
Find all inflection points 7

8. Sec 4.3: Concavity and the Second Derivative Test
Example:
Find all inflection points of

9. Sec 4.3: Concavity and the Second Derivative Test

10. Sec 4.3: Concavity and the Second Derivative Test

11. Sec 4.3: Concavity and the Second Derivative Test1 1 2 2
second Derivative Test: 1 1 2 2
second Derivative Test: 1 1
the test fails.
The function ƒ may have a local maximum, a local minimum, or neither. 2 2

12. Sec 4.3: Concavity and the Second Derivative Test
Critical points:
Local max
Local min

13. Sec 4.3: Concavity and the Second Derivative Test

14. Sec 4.3: Concavity and the Second Derivative Test

15. Sec 4.3: Concavity and the Second Derivative Test
How to find local max and local min
Find all critical points
1st deriv test
2ed deriv test
1st deriv test
No test
Graph or others