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

2. 1 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 Sec 4.3: Concavity and the Second Derivative Test

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

4. F092 Sec 4.3: Concavity and the Second Derivative Test

5. 1 2 Inflection point: + _ _ + Sec 4.3: Concavity and the Second Derivative Test

6. Find all inflection points

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

8. 8 F091 Sec 4.3: Concavity and the Second Derivative Test

10. 1 2 second Derivative Test: 1 2 Sec 4.3: Concavity and the Second Derivative Test 1 2 second Derivative Test: the test fails. The function ƒ may have a local maximum, a local minimum, or neither. 1 2 1 2 1 2

11. F121 Sec 4.3: Concavity and the Second Derivative Test

12. F081 Sec 4.3: Concavity and the Second Derivative Test