Concavity and Inflection Points

1. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

2. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases

3. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( - ) value to a ( + ) value…

4. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( - ) value to a ( + ) value… Concave down – this occurs as the slope of the tangent line decreases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( + ) value to a ( - ) value…

5. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases Concave down – this occurs as the slope of the tangent line decreases

6. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases Concave down – this occurs as the slope of the tangent line decreases

7. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – occur where a graph changes from concave up to concave down, or vice versa

8. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – occur where a graph changes from concave up to concave down, or vice versa

9. Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – will occur where a graph changes from concave up to concave down, or vice versa Inflection point

10. Concavity and Inflection Points

11. Concavity and Inflection Points

12. Concavity and Inflection Points

13. Concavity and Inflection Points 1 0 2. Set up your intervals

14. Concavity and Inflection Points 1 0 2. Set up your intervals

15. Concavity and Inflection Points 2. Set up your intervals

16. Concavity and Inflection Points 2. Set up your intervals - place on number line

17. Concavity and Inflection Points 2. Set up your intervals

18. Concavity and Inflection Points CU 2. Set up your intervals Concave up

19. Concavity and Inflection Points CU CU 2. Set up your intervals Concave up Concave up

20. Concavity and Inflection Points CU CD CU 2. Set up your intervals Concave up Concave up Concave down

21. Concavity and Inflection Points CU CD CU CD 2. Set up your intervals Concave up Concave up Concave down Concave down

22. Concavity and Inflection Points

23. Concavity and Inflection Points

24. Concavity and Inflection Points 2. Set up your intervals

25. Concavity and Inflection Points 2. Set up your intervals

26. Concavity and Inflection Points 2. Set up your intervals - place on number line ** yes, it is possible that a critical point will also be an inflection point

27. Concavity and Inflection Points 2. Set up your intervals

28. Concavity and Inflection Points CU 2. Set up your intervals Concave up

29. Concavity and Inflection Points CU CD 2. Set up your intervals Concave up Concave down

30. Concavity and Inflection Points CU CD CU 2. Set up your intervals Concave up Concave down Concave up

31. Concavity and Inflection Points CU CD CU CU 2. Set up your intervals Concave up Concave down Concave up Concave up