1. II. differentiable at x = 0 III. absolute minimum at x = 0
Warm-Up
Let f be the function given by f(x) = | x |. Which of the following statements are true?
I. continuous at x = 0
II. differentiable at x = 0
III. absolute minimum at x = 0
A) I only C) III only E) II and III only
B) II only D) I and III only

2. Problem of the Day (Calculator)
Let f be the function given by f(x) = | x |. Which of the following statements are true?
I. continuous at x = 0
II. differentiable at x = 0
III. absolute minimum at x = 0
A) I only C) III only E) II and III only
B) II only D) I and III only

3. 3-1:Extrema On An Interval
Objectives: (review)
Find extreme values (maximums and minimums) of a function
Find and use critical numbers

4. Relevant Questions Does f(x) have a maximum value on an interval?
Does f(x) have a minimum value?

5. Important Idea
When we know how to find a function’s extreme values, we can answer questions as:
What is the best strength for medicine given a patient?
What is the least expensive way to manufacture motors?

6. Definition Let f be defined on an interval I containing a:
1. f(a) is a minimum if f(a) f(x) for all x in I.
2. f(a) is a maximum if f(a) f(x) for all x in I.

7. Why does f(x)=x2+1 not have a maximum on the interval (-1,2)?

8. Example
No Maximum
Minimum -1 2
f(x)=x2+1
Open Interval (-1,2)

9. Example
Maximum
Minimum -1 2
f(x)=x2+1
Closed Interval [-1,2]

10. Example
Maximum
No Minimum -1 2
Closed Interval [-1,2]

11. Important Idea
Continuity, or the lack of it, on an interval can affect the existence of extrema on the interval.

12. Extreme Value Theorem
If f is continuous on a closed interval, then f has a both a minimum and a maximum on the interval.

13. Try This
Name a continuous function that over the interval [-10,10] has a maximum equal to its minimum.

14. Definition
Relative (local) extrema is the maximum or minimum over an open interval.
Absolute (global) extrema is the maximum or minimum over a closed interval.

15. Definition
Extrema that occur at endpoints are called endpoint extrema.

16. Important Idea To find absolute extrema: Find the relative extrema
Find the endpoint extrema
Choose the larger or smaller of the values

17. Find the relative and absolute extrema over the interval [-10,10]:
Try This
The absolute extrema are not the same as the relative extrema

18. Find the relative and absolute extrema over the interval [-10,10]:
Try This
The relative and absolute extrema are the same.

19. Try This
Find the value of the derivative at each of the relative extrema shown:
(0,0)
(2,-4)

20. Try This
Find the value of the derivative at each of the relative extrema shown:
(0,0)

21. Definition-Write this down.
Let f be defined at a. If f’(a)=0 or if f ’ is undefined at a, then a is a critical number of f and f(a) is a critical value of f.

22. and this too Relative extrema occur only at critical numbers.
Critical numbers do not guarantee relative extrema

23. Example Find extrema of on the interval [-1,2] Steps:
1. Find critical numbers
2. Evaluate f(x) at each critical number and endpoint
3. Choose max and/or min

24. Try This Find extrema of on the interval [-1,3]
min at (-1,-5);max at (0,0)

25. Lesson Close
Without looking at your notes, name the three steps for finding absolute extrema.

26. Assignment
Page 169 Problems 1-33 odd