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7. Sketch the level curves f(x, y) = c of f (x, y) = x² + y² for c = 1, 2, 4, 6, 9, and 10. In the same coordinate system, sketch the graph of the constraint (x – 1)² + y² = 4. %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to confirm your geometric reasoning. 8. Sketch the level curves f(x, y) = c of f(x, y)= 2x – y for c = -3, –2, -1,0, 1, 2, and 3. In the same coordinate system, sketch the graph of the constraint x² + y² = 1. %3D %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to obtain the desired constrained minimum and maximum algebraically. 9. State the problem in Example 4.26 in Section 4.3 as a constrained optimization problem and solve it using Lagrange multipliers. 10. Explain why it does not make much sense to develop the Lagrange multipliers method to optimize a function f (x, y) of two variables subject to two constraints g1(x, y) = k, and g2(x, y) = k2. Exercises 11 to 19: Find the extreme values (if any) of a function f subject to the given constraint. 11. f(x, y)= 3xy; x² + y² = 4 12. f(x, y)=4-x- y; y – 2x = 1 (Hint: Find the maximum; argue that a minimum does not exist.) %3D

7. Sketch the level curves f(x, y) = c of f (x, y) = x² + y² for c = 1, 2, 4, 6, 9, and 10. In the same coordinate system, sketch the graph of the constraint (x – 1)² + y² = 4. %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to confirm your geometric reasoning. 8. Sketch the level curves f(x, y) = c of f(x, y)= 2x – y for c = -3, –2, -1,0, 1, 2, and 3. In the same coordinate system, sketch the graph of the constraint x² + y² = 1. %3D %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to obtain the desired constrained minimum and maximum algebraically. 9. State the problem in Example 4.26 in Section 4.3 as a constrained optimization problem and solve it using Lagrange multipliers. 10. Explain why it does not make much sense to develop the Lagrange multipliers method to optimize a function f (x, y) of two variables subject to two constraints g1(x, y) = k, and g2(x, y) = k2. Exercises 11 to 19: Find the extreme values (if any) of a function f subject to the given constraint. 11. f(x, y)= 3xy; x² + y² = 4 12. f(x, y)=4-x- y; y – 2x = 1 (Hint: Find the maximum; argue that a minimum does not exist.) %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 17EQ
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7. Sketch the level curves f(x, y) = c of f (x, y) = x² + y² for c = 1, 2, 4, 6, 9, and 10. In the same coordinate system, sketch the graph of the constraint (x – 1)² + y² = 4. %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to confirm your geometric reasoning. 8. Sketch the level curves f(x, y) = c of f(x, y)= 2x – y for c = -3, –2, -1,0, 1, 2, and 3. In the same coordinate system, sketch the graph of the constraint x² + y² = 1. %3D %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to obtain the desired constrained minimum and maximum algebraically. 9. State the problem in Example 4.26 in Section 4.3 as a constrained optimization problem and solve it using Lagrange multipliers. 10. Explain why it does not make much sense to develop the Lagrange multipliers method to optimize a function f (x, y) of two variables subject to two constraints g1(x, y) = k, and g2(x, y) = k2. Exercises 11 to 19: Find the extreme values (if any) of a function f subject to the given constraint. 11. f(x, y)= 3xy; x² + y² = 4 12. f(x, y)=4-x- y; y – 2x = 1 (Hint: Find the maximum; argue that a minimum does not exist.) %3D
Transcribed Image Text:7. Sketch the level curves f(x, y) = c of f (x, y) = x² + y² for c = 1, 2, 4, 6, 9, and 10. In the same coordinate system, sketch the graph of the constraint (x – 1)² + y² = 4. %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to confirm your geometric reasoning. 8. Sketch the level curves f(x, y) = c of f(x, y)= 2x – y for c = -3, –2, -1,0, 1, 2, and 3. In the same coordinate system, sketch the graph of the constraint x² + y² = 1. %3D %3D (a) Looking at your picture, identify the minimum and maximum of f subject to the given constraint. (b) Use the method of Lagrange multipliers to obtain the desired constrained minimum and maximum algebraically. 9. State the problem in Example 4.26 in Section 4.3 as a constrained optimization problem and solve it using Lagrange multipliers. 10. Explain why it does not make much sense to develop the Lagrange multipliers method to optimize a function f (x, y) of two variables subject to two constraints g1(x, y) = k, and g2(x, y) = k2. Exercises 11 to 19: Find the extreme values (if any) of a function f subject to the given constraint. 11. f(x, y)= 3xy; x² + y² = 4 12. f(x, y)=4-x- y; y – 2x = 1 (Hint: Find the maximum; argue that a minimum does not exist.) %3D
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