1. Section 4.4 The Shape of a Graph
MAT 1234 Calculus I
Section 4.4
The Shape of a Graph

2. HW and ….
WebAssign HW
Quiz: 4.3, 4.4
Take time to study for exam 2

3. The 1st Derv. Test Find the critical numbers
Find the intervals of increasing and decreasing
Determine the local max./min.

4. The 1st Derv. Test Find the critical numbers
Find the intervals of increasing and decreasing
Determine the local max./min. Note that intervals of increasing and decreasing are part of the 1st test.

5. The 2nd Derv. Test We will talk about intervals of concave up and down
But they are not part of the 2nd test.

6. Preview We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?

7. Preview We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?
30 second summary!

8. Concave Down 𝑓”(𝑐)<0
Preview
Concave Up 𝑓”(𝑐)>0
Concave Down 𝑓”(𝑐)<0
𝑓’(𝑐)=0
𝑓’(𝑐)=0

9. Preview We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?
30 second summary!
We are going to develop the theory carefully so that it works for all the functions that we are interested in.

10. Preview 4.3 Part I Increasing/Decreasing Test
The First Derivative Test
4.4
Concavity Test
The Second Derivative Test

11. Definition
(a) A function 𝑓 is called concave upward on an interval 𝐼 if the graph of 𝑓 lies above all of its tangents on 𝐼.
(b) A function 𝑓 is called concave downward on an interval 𝐼 if the graph of 𝑓 lies below all of its tangents on 𝐼.

12. Concavity
𝑓 is concave up on 𝐼
Potential local min.

13. Concavity
𝑓 is concave down on 𝐼
Potential local max.

14. Concavity 𝑓 has no local max. or min. 𝑓 has an inflection point at 𝑥=𝑐
Concave down
Concave up
𝑓 has no local max. or min.
𝑓 has an inflection point at 𝑥=𝑐

15. Definition
An inflection point is a point where the concavity changes (from up to down or from down to up)

16. Concavity Test
(a) If 𝑓 ’’(𝑥)>0 on an interval 𝐼, then 𝑓 is concave upward on 𝐼.
(b) If 𝑓 ’’(𝑥)<0 on an interval 𝐼, then 𝑓 is concave downward on 𝐼.

17. Concavity Test
(a) If 𝑓 ’’(𝑥)>0 on an interval 𝐼, then 𝑓 is concave upward on 𝐼.
(b) If 𝑓 ’’(𝑥)<0 on an interval 𝐼, then 𝑓 is concave downward on 𝐼.
Why?

18. Why? 𝑓 ”(𝑥)>0 implies 𝑓’(𝑥) is increasing.
i.e. the slope of tangent lines is increasing.

19. Why? 𝑓 ”(𝑥)<0 implies 𝑓’(𝑥) is decreasing.
i.e. the slope of tangent lines is decreasing.

20. Example 1
Find the intervals of concavity and the inflection points

21. Example 1 Find the intervals of concavity and the inflection points
This is very similar to finding intervals of increasing/decreasing. Instead of looking for 𝑓 ′ 𝑥 =0, we use 𝑓" 𝑥 =0

22. Example 1
This is very similar to finding intervals of increasing/decreasing. Instead of looking for 𝑓 ′ 𝑥 =0, we use 𝑓" 𝑥 =0 𝑥 𝑎 𝑐 𝑏

23. Example 1
This is very similar to finding intervals of increasing/decreasing. Instead of looking for 𝑓 ′ 𝑥 =0, we use 𝑓" 𝑥 =0 𝑥 𝑎 𝑐 𝑏

24. Example 1
(a) Find ,
and the values of such that

25. Example 1
(b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.

26. Example 1 (c) Find the intervals of concavity and inflection point(s).
𝑓( )=
𝑓 has an inflection point at ( , )

27. Expectation Answer in full sentence.
The inflection point should be given by the (𝑥,𝑦) point notation.

28. Example 1 Verification

29. The Second Derivative Test
Suppose 𝑓’’ is continuous near 𝑐.
(a) If 𝑓’(𝑐)=0 and 𝑓’’(𝑐)>0, then 𝑓 has a local minimum at 𝑐.
(b) If 𝑓’(𝑐)=0 and 𝑓’’(𝑐)<0, then f has a local maximum at c.
(c) If 𝑓’’(𝑐)=0, then no conclusion (use 1st derivative test)

30. Second Derivative Test
Suppose
If then 𝑓 has a local min. at 𝑥=𝑐 𝑐
𝑓”(𝑐)>0
𝑓’(𝑐)=0

31. Second Derivative Test
Suppose
If then 𝑓 has a local max. at 𝑥=𝑐 𝑐
𝑓”(𝑐)<0
𝑓’(𝑐)=0

32. Example 2 (Revisit)
Use the second derivative test to find the local max. and local min.

33. Example 2 (Revisit)
(a) Find the critical numbers of

34. Example 2 (Revisit)
(b) Use the Second Derivative Test to find the local max/min of
The local max. value of 𝑓 is
The local min. value of 𝑓 is

35. Second Derivative Test
Step 1: Find the critical points
Step 2: For each critical point,
determine the sign of the second derivative;
Find the function value
Make a formal conclusion
Note that no other steps are required such as finding intervals of inc/dec, concave up/down.

36. The Second Derivative Test
(c) If 𝑓’’(𝑐)=0, then no conclusion

37. The Second Derivative Test
(c) If 𝑓’’(𝑐)=0, then no conclusion

38. The Second Derivative Test
(c) If 𝑓’’(𝑐)=0, then no conclusion

39. The Second Derivative Test
(c) If 𝑓’’(𝑐)=0, then no conclusion

40. The Second Derivative Test
Suppose 𝑓’’ is continuous near 𝑐.
(a) If 𝑓’(𝑐)=0 and 𝑓’’(𝑐)>0, then 𝑓 has a local minimum at 𝑐.
(b) If 𝑓’(𝑐)=0 and 𝑓’’(𝑐)<0, then f has a local maximum at c.
(c) If 𝑓’’(𝑐)=0, then no conclusion (use 1st derivative test)

41. Which Test is Easier?
First Derivative Test
Second Derivative Test

42. Final Reminder
You need intervals of increasing/decreasing for the First Derivative Test.
You do not need intervals of concavity for the Second Derivative Test.