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Chapter 8 Multivariable Calculus Section 3 Maxima and Minima (Part 1)
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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.
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3 Barnett/Ziegler/Byleen Business Calculus 12e Local Maxima and Minima 2D Local extrema 3D Local extrema
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4 Review Barnett/Ziegler/Byleen Business Calculus 12e
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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
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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
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7 Barnett/Ziegler/Byleen Business Calculus 12e Second-Derivative Test for Local Extrema
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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
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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):
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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).
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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.)
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12 Barnett/Ziegler/Byleen Business Calculus 12e Example 2
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13 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (cont.) Since AC – B 2 < 0 then f (0, 0)=3 is a saddle point.
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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.
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15 Barnett/Ziegler/Byleen Business Calculus 12e Example 3
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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.
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17 Example 4 Barnett/Ziegler/Byleen Business Calculus 12e
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18 Example 5 Barnett/Ziegler/Byleen Business Calculus 12e
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19 Homework Barnett/Ziegler/Byleen Business Calculus 12e
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Chapter 8 Multivariable Calculus Section 3 Maxima and Minima (Part 2)
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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.
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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
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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))
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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
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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
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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.
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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
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28 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e
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29 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Now find critical point of P(x,y): Critical point is (5,3)
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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)
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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
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32 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Material = Base + front & back + 2 sides & partition This is the function we must minimize.
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33 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e First find critical point of M(x,y): Critical point is (6,4)
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34 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e Critical point is (6,4) Now find A, B & C:
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35 Homework Barnett/Ziegler/Byleen Business Calculus 12e
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