1. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 1

2. 6.3 Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 2 Maximum-Minimum Problems OBJECTIVE Find relative extrema of a function of two variables.

3. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 3 DEFINITION: A function f of two variables: 1. has a relative maximum at (a, b) if f (x, y) ≤ f (a, b) for all points (x, y) in a region containing (a, b). 2. has a relative minimum at (a, b) if f (x, y) ≥ f (a, b) for all points (x, y) in a region containing (a, b). 6.3 Maximum-Minimum Problems

4. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 4 THEOREM 1: The D-Test If f is a differentiable function of x and y, to find the relative maximum and minimum values of f: 1. Find f x, f y, f xx, f yy, and f xy. 2. Solve the system of equations f x = 0, f y = 0. Let (a, b) represent a solution. 3. Evaluate D, where D = f xx (a, b)·f yy (a, b) – [ f xy (a, b)] 2. 6.3 Maximum-Minimum Problems

5. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 5 THEOREM 1 (concluded): 4. Then: a) f has a maximum at (a, b) if D > 0 and f xx (a, b) < 0. b) f has a minimum at (a, b) if D > 0 and f xx (a, b) > 0. c) f has neither a maximum nor a minimum at (a, b) if D < 0. The function has a saddle point at (a, b). d) This test is not applicable if D = 0. 6.3 Maximum-Minimum Problems

6. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 6 Example 1: Find the relative maximum or minimum values of 1. Find f x, f y, f xx, f yy, and f xy. 6.3 Maximum-Minimum Problems

7. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 7 Example 1 (continued): 2. Solve the system of equations f x = 0 and f y = 0. Using substitution, 6.3 Maximum-Minimum Problems

8. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 8 Example 1 (continued): Then, substituting back, Thus, (2, –1) is the only critical point. 3. Find D. 6.3 Maximum-Minimum Problems

9. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 9 Example 1 (concluded): 4. Since D = 3 and f xx (2, –1) = 2, since D > 0 and f xx (2, –1) > 0, if follows from the D-Test that f has a relative minimum at (2, –1). The minimum value is 6.3 Maximum-Minimum Problems

10. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 10 6.3 Maximum-Minimum Problems Quick Check 1 Find the relative maximum and minimum values of 1. Find 2. Solve the system of equations Thus is the only critical point.

11. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 11 6.3 Maximum-Minimum Problems Quick Check 1 Concluded 3. Find D. 4. Since it follows from the D-Test that f has a minimum at. The minimum value is

12. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 12 Example 2: Find the relative maximum and minimum values of 1. Find f x, f y, f xx, f yy, and f xy. 6.3 Maximum-Minimum Problems

13. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 13 Example 2 (continued): 2. Solve the system of equations f x = 0 and f y = 0. Using substitution, 6.3 Maximum-Minimum Problems

14. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 14 Example 2 (continued): Then, substitute back to find y for both values of x. Thus, (0,0) and (1/6, 1/12) are two critical points. 6.3 Maximum-Minimum Problems

15. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 15 Example 2 (continued): 3. Find D for both critical points. First, for (0,0) Then, for (1/6, 1/12) 6.3 Maximum-Minimum Problems

16. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 16 Example 2 (concluded): 4. Since, for (0,0), D < 0, (0,0) is neither a maximum nor a minimum value but a saddle point. For (1/6, 1/12), D > 0 and f xx (1/6, 1/12) < 0. Therefore, f has a relative maximum at (1/6, 1/12). 6.3 Maximum-Minimum Problems

17. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 17 6.3 Maximum-Minimum Problems Quick Check 2 Find the critical points of Then use the D-test to classify each point as a relative maximum, a relative minimum, or a saddle point. 1.

18. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 18 6.3 Maximum-Minimum Problems Quick Check 2 Continued 2. Solve for the system of equations Thus there are critical points at

19. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 19 6.3 Maximum-Minimum Problems Quick Check 2 Continued 3. Find D for both critical points. First for Then, for

20. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 20 6.3 Maximum-Minimum Problems Quick Check 2 Concluded 4. Since, for is neither a minimum nor a maximum value, but is a saddle point. Therefore there is a relative maximum at The relative maximum is

21. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 21 Example 3: A firm produces two kinds of golf balls, one that sells for $3 and one priced at $2. The total revenue, in thousands of dollars, from the sale of x thousand balls at $3 each and y thousand at $2 each is given by The company determines that the total cost, in thousands of dollars, of producing x thousand of the $3 ball and y thousand of the $2 ball is given by 6.3 Maximum-Minimum Problems

22. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 22 Example 3 (continued): How many balls of each type must be produced and sold in order to maximize profit? The total profit function P(x,y) is given by 6.3 Maximum-Minimum Problems

23. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 23 Example 3 (continued): 1. Find P x, P y, P xx, P yy, and P xy. 2. Solve the system of equations P x = 0 and P y = 0. 6.3 Maximum-Minimum Problems

24. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 24 Example 3 (continued): Adding these two equations, we get Then, Now, substitute back into P x = 0 or P y = 0 to find y. 6.3 Maximum-Minimum Problems

25. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 25 Example 3 (continued): Thus, (4, 2) is the only critical point. 3. Find D. 6.3 Maximum-Minimum Problems

26. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 26 Example 3 (continued): Thus, since D > 0 and P xx (4,2) < 0, it follows that P has a relative maximum at (4,2). So in order to maximize profit, the company must produce and sell 4 thousand of the $3 golf ball and 2 thousand of the $2 golf ball. The maximum profit will be 6.3 Maximum-Minimum Problems

27. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 27 6.3 Maximum-Minimum Problems Quick Check 3 Repeat Example 3 using the same cost function and assuming that the company’s total revenue, in thousands of dollars, comes from the sale of x thousand balls at $3.50 each and y thousand at $3.75 each. The total profit function is given by

28. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 28 6.3 Maximum-Minimum Problems Quick Check 3 Continued 1. 2. Solve the system of equations Add the two equations together and you get Substitute this into one of the equations and you get Thus the only critical value is at

29. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 29 6.3 Maximum-Minimum Problems Quick Check 3 Concluded 3. Find D. 4. Thus, since P has a relative maximum at The maximum is Thus, the maximum profit is $17.031 thousand when x = $4.625 thousand and y = $3 thousand.

30. Copyright © 2016, 2012 Pearson Education, Inc. 6.3 - 30 6.3 Maximum-Minimum Problems Section Summary A two-variable function f has a relative maximum at if for all points in a region containing and has a relative minimum at if for all points in a region containing The D-test is used to classify a critical point as a relative minimum, a relative maximum, or a saddle point.