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Find the extreme values of f subject to both constraints. z)=r+ v + z: x² + z² = 2, x + y = 1

Find the extreme values of f subject to both constraints. z)=r+ v + z: x² + z² = 2, x + y = 1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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20 YA 0 29 м and so μ² = 22, μ = 70 9. ⁹. f(x, y, z) = xy²z; z = 1 = x + y = 1 ± 7/√29. The corresponding values of f are 14.8 EXERCISES g(x, y) = 8. Estimate the maximum and minimum values 1. Pictured are a contour map of f and a curve with equation off subject to the constraint that g(x, y) reasoning. = 8. Explain your √/29 + 2( = √/29) + 3(1 = Therefore the maximum value of f on the given curve is 3 + √29 g(x, y) = 8 60 50 40 30 20 10 x² + y² = 1 x² + y² = 10 X 3. f(x, y) = x² - y²; 4. f(x, y) = 3x + y; 5. f(x, y) = xy; 4x² + y² = 8 6. f(x,y) = xe”; x + y = 2 7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9 8. f(x, y, z) = exy²; 2x² + y² + z² = 24 x² + y² + z² = 4 It + 2 2. (a) Use a graphing calculator or computer to graph the circle x² + y² = 1. On the same screen, graph several curves of the form x² + y = c until you find two that just touch the circle. What is the significance of the values of c for these two curves? ± √29/2. Then x = ± √29/2. Then x = 2/√√29, y = ±5/√29, and, from (20), (b) Use Lagrange multipliers to find the extreme values of f(x, y) = x² + y subject to the constraint x² + y² = 1. Compare your answers with those in part (a). D 3-14 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 01 10. f(x, y, z) = ln(x² + 1) + ln(y² + 1) + ln(z² + 1); x² + y² + z² = 12 7 11. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 12. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 13. f(x, y, z, t) = x+y+z+t; x² + y² + z² + t² = 1 14. f(x1, x2,...,xn) = x₁ + x₂ + x₂ + ... + xn; x² + x² + + x² = 1 x₁ + ... = 29 = 3 ± √29 15. The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of x² + y² subject to the constraint xy f(x, y) 1 can be solved using Lagrange multipliers, but f does not have a maximum value with that constraint. DE = 16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z² subject to the constraint x + 2y + 3z = 10. Show that f has no maximum value with this constraint. qu 17-20 Find the extreme values of f subject to both constraints. 17. f(x, y, z) = x+y+z; x² + z² = 2, x + y = 1 18. f(x, y, z) = z; x² + y² = z², x+y+z=24 19. f(x, y, z) = yz + xy; xy = 1, y² + z² = 1 20. f(x, y, z) = x² + y² + z²; x - y = 1, y² = z² = 1 - 21-23 Find the extreme values of f on the region described by the inequality. 21. f(x, y) = x² + y² + 4x - 4y, x² + y² ≤ 9 22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16 23. f(x, y) = exy, x² + 4y² ≤ 1 GENE X 24. Consider the problem of maximizing the function f(x, y) = 2x + 3y subject to the constraint √√x + √√y = 5. (a) Try using Lagrange multipliers to solve the problem. (b) Does f(25, 0) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint equation and several level curves of f. (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of f(9, 4)?
Transcribed Image Text:20 YA 0 29 м and so μ² = 22, μ = 70 9. ⁹. f(x, y, z) = xy²z; z = 1 = x + y = 1 ± 7/√29. The corresponding values of f are 14.8 EXERCISES g(x, y) = 8. Estimate the maximum and minimum values 1. Pictured are a contour map of f and a curve with equation off subject to the constraint that g(x, y) reasoning. = 8. Explain your √/29 + 2( = √/29) + 3(1 = Therefore the maximum value of f on the given curve is 3 + √29 g(x, y) = 8 60 50 40 30 20 10 x² + y² = 1 x² + y² = 10 X 3. f(x, y) = x² - y²; 4. f(x, y) = 3x + y; 5. f(x, y) = xy; 4x² + y² = 8 6. f(x,y) = xe”; x + y = 2 7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9 8. f(x, y, z) = exy²; 2x² + y² + z² = 24 x² + y² + z² = 4 It + 2 2. (a) Use a graphing calculator or computer to graph the circle x² + y² = 1. On the same screen, graph several curves of the form x² + y = c until you find two that just touch the circle. What is the significance of the values of c for these two curves? ± √29/2. Then x = ± √29/2. Then x = 2/√√29, y = ±5/√29, and, from (20), (b) Use Lagrange multipliers to find the extreme values of f(x, y) = x² + y subject to the constraint x² + y² = 1. Compare your answers with those in part (a). D 3-14 Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. 01 10. f(x, y, z) = ln(x² + 1) + ln(y² + 1) + ln(z² + 1); x² + y² + z² = 12 7 11. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 12. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 13. f(x, y, z, t) = x+y+z+t; x² + y² + z² + t² = 1 14. f(x1, x2,...,xn) = x₁ + x₂ + x₂ + ... + xn; x² + x² + + x² = 1 x₁ + ... = 29 = 3 ± √29 15. The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of x² + y² subject to the constraint xy f(x, y) 1 can be solved using Lagrange multipliers, but f does not have a maximum value with that constraint. DE = 16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z² subject to the constraint x + 2y + 3z = 10. Show that f has no maximum value with this constraint. qu 17-20 Find the extreme values of f subject to both constraints. 17. f(x, y, z) = x+y+z; x² + z² = 2, x + y = 1 18. f(x, y, z) = z; x² + y² = z², x+y+z=24 19. f(x, y, z) = yz + xy; xy = 1, y² + z² = 1 20. f(x, y, z) = x² + y² + z²; x - y = 1, y² = z² = 1 - 21-23 Find the extreme values of f on the region described by the inequality. 21. f(x, y) = x² + y² + 4x - 4y, x² + y² ≤ 9 22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16 23. f(x, y) = exy, x² + 4y² ≤ 1 GENE X 24. Consider the problem of maximizing the function f(x, y) = 2x + 3y subject to the constraint √√x + √√y = 5. (a) Try using Lagrange multipliers to solve the problem. (b) Does f(25, 0) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint equation and several level curves of f. (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of f(9, 4)?
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