2. 3-1:Extrema On An Interval Objectives : Find extreme values (maximums and minimums) of a function Find and use critical numbers ©2002 Roy L. Gover (www.mrgover.com)

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

4. 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?

5. 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.

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

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

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

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

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

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

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

13. 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.

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

15. Important Idea To find absolute extrema: 1.Find the relative extrema 2.Find the endpoint extrema 3.Choose the larger or smaller of the values

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

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

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

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

20. Definition 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.

21. Important Ideas Relative extrema occur only at critical numbers. Critical numbers do not guarantee relative extrema

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

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

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

25. Assignment Pages 165-6 #7-10, 21,25,31,49,51,53