Chapter 8 Multivariable Calculus Section 3 Maxima and Minima (Part 1)

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.3 Maxima and Minima The student will be able to identify critical points and maxima and minima of functions.

3 Barnett/Ziegler/Byleen Business Calculus 12e Local Maxima and Minima 2D Local extrema 3D Local extrema

4 Review Barnett/Ziegler/Byleen Business Calculus 12e

5 3D Local Extrema  We will assume that the 3D surface is smooth and has no sharp points or edges.  We will only be concerned about local extrema, not absolute extrema. Barnett/Ziegler/Byleen Business Calculus 12e

6 3D Local Extrema Theorem 1: Let f (a, b) be a local extremum (local minimum or maximum) for the function f. Then f x (a, b) = 0 and f y (a, b) = 0

7 Barnett/Ziegler/Byleen Business Calculus 12e Second-Derivative Test for Local Extrema

8 Barnett/Ziegler/Byleen Business Calculus 12e 2 nd Derivative Test (continued) Case 1. f (a, b) is a local maximum if AC – B 2 > 0 and A < 0 Case 2. f (a, b) is a local minimum if AC – B 2 > 0 and A > 0 Case 3. f (a, b) is a saddle point if AC – B 2 < 0 Case 4. inconclusive if AC – B 2 = 0 Local Minimum Local Maximum Saddle Point

9 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Find local extrema for f (x, y) = 3 – x 2 – y 2 + 6y Step 1. Find the critical point(s):

10 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 (cont.) Find local extrema for f (x, y) = 3 – x 2 – y 2 + 6y Step 2. Compute A = f xx (0, 3), B = f xy (0, 3), C = f yy (0, 3).

11 Barnett/Ziegler/Byleen Business Calculus 12e Example 1: Since AC – B 2 > 0 and A < 0, then f (0, 3)=12 is a local maximum. Example 1 (cont.)

12 Barnett/Ziegler/Byleen Business Calculus 12e Example 2

13 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (cont.) Since AC – B 2 < 0 then f (0, 0)=3 is a saddle point.

14 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (cont.) Since AC – B 2 > 0 and A>0 then f (2, 2)=-8 is a local minimum.

15 Barnett/Ziegler/Byleen Business Calculus 12e Example 3

16 Barnett/Ziegler/Byleen Business Calculus 12e Example 3 (continued) Since AC – B 2 > 0 and A < 0 then f (2, 3)=11 is a local maximum.

17 Example 4 Barnett/Ziegler/Byleen Business Calculus 12e

18 Example 5 Barnett/Ziegler/Byleen Business Calculus 12e

19 Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 8 Multivariable Calculus Section 3 Maxima and Minima (Part 2)

21 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.3 Maxima and Minima The student will be able to solve applications by finding critical points and maxima and minima of functions.

22 Review Barnett/Ziegler/Byleen Business Calculus 12e Case 1. f (a, b) is a local maximum if AC – B 2 > 0 and A < 0 Case 2. f (a, b) is a local minimum if AC – B 2 > 0 and A > 0 Case 3. f (a, b) is a saddle point if AC – B 2 < 0 Case 4. inconclusive if AC – B 2 = 0

23 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 The annual labor and automated equipment cost (in millions of dollars) for producing TV sets is given by C (x, y) = 2x 2 + 2xy + 3y 2 – 16x – 18y + 54, where x is the amount spent per year on labor, and y is the amount spent per year on automated equipment (both in millions of dollars). Minimize the cost (find the minimum of C(x,y))

24 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 (continued) C (x, y) = 2x 2 + 2xy + 3y 2 – 16x – 18y + 54 Step 1. Find the critical point: C x ( x, y) = C y ( x, y) = 4x + 2y – 16 = 0 2x + 6y – 18 = 0 4x + 2y – 16 2x + 6y – 18

25 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 (continued) Step 2. Compute A = C xx (3, 2), B = C xy (3, 2), C = C yy (3, 2). C x (x, y) = 4x + 2y – 16 C y (x, y) = 2x + 6y – 18 C (x, y) = 2x 2 + 2xy + 3y 2 – 16x – 18y + 54

26 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 (continued) Step 3. Evaluate AC – B 2 and classify the critical point. AC – B 2 = 20 Since AC – B 2 > 0 and A > 0, then f(3, 2) = 12 is a local minimum The minimum cost is 12 million dollars. It occurs when they spend 3 million dollars on labor and 2 million dollars on automated equipment per year.

27 Example 2  A satellite TV station is to be located at P(x,y) so that the sum of the squares of the distances from P to the three towns A, B, and C is minimized. Find the coordinates of P that will minimize the cost of providing satellite TV for all three towns. Barnett/Ziegler/Byleen Business Calculus 12e

28 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

29 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Now find critical point of P(x,y): Critical point is (5,3)

30 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued) Critical point is (5,3) Now find A, B & C: P(5,3)=96 is a local minimum. The coordinates of the location of the TV satellite that will minimize the cost of providing services to all 3 towns is: P(5,3)

31 Example 3  The packaging department in a company needs to design a rectangular box with no top and a partition down the middle.  The box must have a volume of 48 cubic inches.  Find the dimensions of the box that will minimize the amount of material used to construct the box.  How much material will be used? Barnett/Ziegler/Byleen Business Calculus 12e

32 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Material = Base + front & back + 2 sides & partition This is the function we must minimize.

33 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e First find critical point of M(x,y): Critical point is (6,4)

34 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Critical point is (6,4) Now find A, B & C:

35 Homework Barnett/Ziegler/Byleen Business Calculus 12e