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

13. 2. Set up your intervals 0 1

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

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 2. Set up your intervals Concave up CU

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

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

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

22. Concavity and Inflection Points

24. 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 2. Set up your intervals CU Concave up

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

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

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