1. 3.2 – Concavity and Points of Inflection
Math 140
3.2 – Concavity and Points of Inflection

2. The 2nd derivative tells us how the 1st derivative is changing
The 2nd derivative tells us how the 1st derivative is changing. If 𝑓 ′′ is positive, then 𝑓 ′ is _________________, and the graph of 𝑓 is ____________________. If 𝑓 ′′ is negative, then 𝑓 ′ is ________________, and the graph of 𝑓 is ___________________.

3. The 2nd derivative tells us how the 1st derivative is changing
The 2nd derivative tells us how the 1st derivative is changing. If 𝑓 ′′ is positive, then 𝑓 ′ is _________________, and the graph of 𝑓 is ____________________. If 𝑓 ′′ is negative, then 𝑓 ′ is ________________, and the graph of 𝑓 is ___________________.
increasing

4. The 2nd derivative tells us how the 1st derivative is changing
The 2nd derivative tells us how the 1st derivative is changing. If 𝑓 ′′ is positive, then 𝑓 ′ is _________________, and the graph of 𝑓 is ____________________. If 𝑓 ′′ is negative, then 𝑓 ′ is ________________, and the graph of 𝑓 is ___________________.
increasing
concave up

5. The 2nd derivative tells us how the 1st derivative is changing
The 2nd derivative tells us how the 1st derivative is changing. If 𝑓 ′′ is positive, then 𝑓 ′ is _________________, and the graph of 𝑓 is ____________________. If 𝑓 ′′ is negative, then 𝑓 ′ is ________________, and the graph of 𝑓 is ___________________.
increasing
concave up
decreasing

6. The 2nd derivative tells us how the 1st derivative is changing
The 2nd derivative tells us how the 1st derivative is changing. If 𝑓 ′′ is positive, then 𝑓 ′ is _________________, and the graph of 𝑓 is ____________________. If 𝑓 ′′ is negative, then 𝑓 ′ is ________________, and the graph of 𝑓 is ___________________.
increasing
concave up
decreasing
concave down

8. slopes increasing

9. slopes increasing
slopes decreasing

10. Where might 𝑓(𝑥) change from concave up to concave down, or concave down to concave up? 1. When 𝑓 ′′ 𝑥 =0 2. When 𝑓′′(𝑥) does not exist (DNE)

11. Where might 𝑓(𝑥) change from concave up to concave down, or concave down to concave up? 1. When 𝑓 ′′ 𝑥 =0 2. When 𝑓′′(𝑥) does not exist (DNE)

12. Where might 𝑓(𝑥) change from concave up to concave down, or concave down to concave up? 1. When 𝑓 ′′ 𝑥 =0 2. When 𝑓′′(𝑥) does not exist (DNE)

13. Ex 1. Determine the intervals of concavity for 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −3 𝑥 2 +1.

14. Ex 1. Determine the intervals of concavity for 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −3 𝑥 2 +1.

15. Ex 1. Determine the intervals of concavity for 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −3 𝑥 2 +1.

16. Ex 1. Determine the intervals of concavity for 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −3 𝑥 2 +1.

17. Ex 1. Determine the intervals of concavity for 𝑓 𝑥 = 𝑥 4 + 𝑥 3 −3 𝑥 2 +1.

21. A point 𝑐, 𝑓 𝑐 where the concavity changes is called an _______________________. (Again, this can happen if either 𝑓 ′′ 𝑐 =0 or 𝑓 ′′ 𝑐 does not exist.)

22. A point 𝑐, 𝑓 𝑐 where the concavity changes is called an _______________________. (Again, this can happen if either 𝑓 ′′ 𝑐 =0 or 𝑓 ′′ 𝑐 does not exist.)
inflection point

23. A point 𝑐, 𝑓 𝑐 where the concavity changes is called an _______________________. (Again, this might happen if either 𝑓 ′′ 𝑐 =0 or 𝑓 ′′ 𝑐 does not exist.)
inflection point

29. Ex 2. Find all inflection points of 𝑓 𝑥 =3 𝑥 5 −5 𝑥 4 −1

30. Ex 2. Find all inflection points of 𝑓 𝑥 =3 𝑥 5 −5 𝑥 4 −1

31. Ex 2. Find all inflection points of 𝑓 𝑥 =3 𝑥 5 −5 𝑥 4 −1

32. Ex 3. Find all inflection points of 𝑔 𝑥 = 𝑥 1 3 .

33. Ex 3. Find all inflection points of 𝑔 𝑥 = 𝑥 1 3 .

34. Ex 3. Find all inflection points of 𝑔 𝑥 = 𝑥 1 3 .

35. 2nd Derivative Test
Suppose 𝑓 ′ 𝑐 =0. If 𝑓 ′′ 𝑐 >0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 <0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 =0 or 𝑓′′(𝑐) does not exist, then test doesn’t say anything (maybe try 1st Derivative Test).

36. 2nd Derivative Test
Suppose 𝑓 ′ 𝑐 =0. If 𝑓 ′′ 𝑐 >0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 <0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 =0 or 𝑓′′(𝑐) does not exist, then test doesn’t say anything (maybe try 1st Derivative Test).

37. 2nd Derivative Test minimum
Suppose 𝑓 ′ 𝑐 =0. If 𝑓 ′′ 𝑐 >0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 <0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 =0 or 𝑓′′(𝑐) does not exist, then test doesn’t say anything (maybe try 1st Derivative Test).
minimum

38. 2nd Derivative Test minimum
Suppose 𝑓 ′ 𝑐 =0. If 𝑓 ′′ 𝑐 >0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 <0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 =0 or 𝑓′′(𝑐) does not exist, then test doesn’t say anything (maybe try 1st Derivative Test).
minimum

39. 2nd Derivative Test minimum maximum
Suppose 𝑓 ′ 𝑐 =0. If 𝑓 ′′ 𝑐 >0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 <0, then there is a relative ________________ at 𝑥=𝑐. If 𝑓 ′′ 𝑐 =0 or 𝑓′′(𝑐) does not exist, then test doesn’t say anything (maybe try 1st Derivative Test).
minimum
maximum

40. Ex 4. Use the 2nd Derivative Test to find the relative maxima and minima of 𝑓 𝑥 =2 𝑥 3 +3 𝑥 2 −12𝑥−7.

41. Ex 4. Use the 2nd Derivative Test to find the relative maxima and minima of 𝑓 𝑥 =2 𝑥 3 +3 𝑥 2 −12𝑥−7.

42. Ex 4. Use the 2nd Derivative Test to find the relative maxima and minima of 𝑓 𝑥 =2 𝑥 3 +3 𝑥 2 −12𝑥−7.

43. Note: 𝑓 𝑥 = 𝑥 4 𝑓 ′ 𝑥 =4 𝑥 3 𝑓 ′′ 𝑥 =12 𝑥 2 Note that here, 𝑓 ′ 𝑥 =0 when 𝑥=0, but 𝑓 ′′ 0 =0, so the 2nd Derivative Test is inconclusive.

44. Summary
1st Derivative Test – Uses sign of 𝒇′ across a critical number to find relative max/min. 2nd Derivative Test – Uses sign of 𝒇 ′′ at a critical number to find relative max/min.