Explanation Given Information: The provided function is f ( x , y ) = x 2 − y 2 subject to the constraint 2 y − x 2 = 0 . Consider that the variables x and y are positive. Formula used: Method of Lagrange multipliers, If the function f ( x , y ) contains a maximum or minimum subject to the constraint g ( x , y ) = 0 then the maximum or minimum can occur at one of the critical numbers of the function F is, F ( x , y , λ ) = f ( x , y ) − λ g ( x , y ) where, λ is a Lagrange multiplier. Steps to determine the minimum or maximum of the function f . 1. Solve the system of equations, F x ( x , y , λ ) = 0 F y ( x , y , λ ) = 0 F λ ( x , y , λ ) = 0 2. Determine the value of the function f at each solution obtained from the step 1. The largest value gives the maximum value of function f subject to the constraint g ( x , y ) = 0 and the lowest value gives the minimum value of function f subject to the constraint g ( x , y ) = 0 . Calculation: Consider the function, f ( x , y ) = x 2 − y 2 The provided constraint is 2 y − x 2 = 0 . So, g ( x , y ) = 2 y − x 2 Now, the new function F is, F ( x , y , λ ) = x 2 − y 2 − λ ( 2 y − x 2 ) Find the partial derivative of function F to determine the critical number for F . The partial derivatives of F with respect to x and then set it to zero. F x ( x , y , λ ) = ∂ ∂ x [ x 2 − y 2 − λ ( 2 y − x 2 ) ] 0 = 2 x + 2 λ x 0 = 2 x ( 1 + λ ) λ = − 1 The value x = 0 does not gives maximum value for the function

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Question

Chapter 7.6, Problem 6E

To determine

To calculate: The maximum value of the function f(x,y)=x2−y2 subject to the constraint 2y−x2=0 by the use of Lagrange multipliers.

Expert Solution & Answer

Chapter 7 Solutions

Calculus: An Applied Approach (MindTap Course List)

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Finding Second Partial Derivatives In Exercises...Ch. 7.4 - Marginal Cost A company manufactures mountain...Ch. 7.4 - Prob. 62E Ch. 7.4 - Prob. 63E Ch. 7.4 - Prob. 64E Ch. 7.4 - Prob. 65E Ch. 7.4 - Prob. 66E Ch. 7.4 - Prob. 67E Ch. 7.4 - Prob. 68E Ch. 7.4 - Prob. 69E Ch. 7.4 - Investment The value of an investment of $ 100...Ch. 7.4 - Prob. 72E Ch. 7.4 - Prob. 73E Ch. 7.5 - Prob. 1CP Ch. 7.5 - Prob. 2CP Ch. 7.5 - Find the relative extrema and saddle points of...Ch. 7.5 - Prob. 4CP Ch. 7.5 - Prob. 5CP Ch. 7.5 - Prob. 1SWU Ch. 7.5 - Prob. 2SWU Ch. 7.5 - Prob. 3SWU Ch. 7.5 - Prob. 4SWU Ch. 7.5 - Prob. 5SWU Ch. 7.5 - Prob. 6SWU Ch. 7.5 - Prob. 7SWU Ch. 7.5 - Prob. 8SWU Ch. 7.5 - Prob. 9SWU Ch. 7.5 - Prob. 10SWU Ch. 7.5 - Prob. 11SWU Ch. 7.5 - Prob. 12SWU Ch. 7.5 - Prob. 13SWU Ch. 7.5 - Prob. 14SWU Ch. 7.5 - Prob. 1E Ch. 7.5 - Prob. 2E Ch. 7.5 - Prob. 3E Ch. 7.5 - Prob. 4E Ch. 7.5 - Prob. 5E Ch. 7.5 - Prob. 6E Ch. 7.5 - Prob. 7E Ch. 7.5 - Prob. 8E Ch. 7.5 - Prob. 9E Ch. 7.5 - Prob. 10E Ch. 7.5 - 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Cost An automobile manufacturer has determined...Ch. 7.5 - Prob. 50E Ch. 7.5 - Biology A lake is to be stocked with smallmouth...Ch. 7.5 - Prob. 53E Ch. 7.5 - Shannon Diversity Index One way to measure species...Ch. 7.5 - Prob. 55E Ch. 7.5 - Prob. 56E Ch. 7.5 - Prob. 1QY Ch. 7.5 - Prob. 2QY Ch. 7.5 - Prob. 3QY Ch. 7.5 - Prob. 4QY Ch. 7.5 - Prob. 5QY Ch. 7.5 - Prob. 6QY Ch. 7.5 - Prob. 7QY Ch. 7.5 - Prob. 8QY Ch. 7.5 - Prob. 9QY Ch. 7.5 - Prob. 10QY Ch. 7.5 - Prob. 11QY Ch. 7.5 - Prob. 12QY Ch. 7.5 - Prob. 13QY Ch. 7.5 - Prob. 14QY Ch. 7.5 - Prob. 15QY Ch. 7.5 - Prob. 16QY Ch. 7.5 - Prob. 17QY Ch. 7.5 - Prob. 18QY Ch. 7.5 - Prob. 19QY Ch. 7.5 - Prob. 20QY Ch. 7.5 - Prob. 21QY Ch. 7.5 - Prob. 22QY Ch. 7.5 - Prob. 23QY Ch. 7.5 - Prob. 24QY Ch. 7.5 - Prob. 25QY Ch. 7.5 - Prob. 26QY Ch. 7.6 - Find the maximum volume of V=xyz subject to the...Ch. 7.6 - Prob. 2CP Ch. 7.6 - Prob. 3CP Ch. 7.6 - Prob. 4CP Ch. 7.6 - Prob. 1SWU Ch. 7.6 - Prob. 2SWU Ch. 7.6 - Prob. 3SWU Ch. 7.6 - Prob. 4SWU Ch. 7.6 - 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Prob. 34E Ch. 7.6 - Prob. 35E Ch. 7.6 - Prob. 36E Ch. 7.6 - Prob. 37E Ch. 7.6 - Least-Cost Rule Repeat Exercise 37 for the...Ch. 7.6 - Construction An animal shelter plans to use the...Ch. 7.6 - Office Space Partitions will be used in an office...Ch. 7.6 - Biology A microbiologist must prepare a culture...Ch. 7.6 - Prob. 42E Ch. 7.6 - Nutrition The number of grams of your favorite ice...Ch. 7.6 - HOW DO YOU SEE IT? The graphs show the constraint...Ch. 7.6 - Prob. 45E Ch. 7.7 - Prob. 1CP Ch. 7.7 - Prob. 2CP Ch. 7.7 - Prob. 3CP Ch. 7.7 - Prob. 1SWU Ch. 7.7 - Prob. 2SWU Ch. 7.7 - Prob. 3SWU Ch. 7.7 - Prob. 4SWU Ch. 7.7 - Prob. 5SWU Ch. 7.7 - Prob. 6SWU Ch. 7.7 - Prob. 7SWU Ch. 7.7 - Prob. 8SWU Ch. 7.7 - Prob. 1E Ch. 7.7 - Prob. 2E Ch. 7.7 - Prob. 3E Ch. 7.7 - Prob. 4E Ch. 7.7 - Prob. 5E Ch. 7.7 - Prob. 6E Ch. 7.7 - Prob. 7E Ch. 7.7 - Prob. 8E Ch. 7.7 - Prob. 9E Ch. 7.7 - Prob. 10E Ch. 7.7 - Prob. 11E Ch. 7.7 - Prob. 12E Ch. 7.7 - Prob. 13E Ch. 7.7 - Prob. 14E Ch. 7.7 - Prob. 15E Ch. 7.7 - Demand A hardware retailer wants to know the...Ch. 7.7 - Prob. 17E Ch. 7.7 - HOW DO YOU SEE IT? Match the regression equation...Ch. 7.7 - Prob. 19E Ch. 7.7 - Prob. 20E Ch. 7.7 - Prob. 21E Ch. 7.7 - Prob. 22E Ch. 7.7 - Prob. 23E Ch. 7.7 - Prob. 24E Ch. 7.7 - Prob. 25E Ch. 7.7 - Prob. 26E Ch. 7.7 - Prob. 27E Ch. 7.7 - Prob. 28E Ch. 7.7 - Prob. 29E Ch. 7.7 - Prob. 30E Ch. 7.8 - Prob. 1CP Ch. 7.8 - Prob. 2CP Ch. 7.8 - Prob. 3CP Ch. 7.8 - Prob. 4CP Ch. 7.8 - Prob. 5CP Ch. 7.8 - Prob. 6CP Ch. 7.8 - Prob. 1SWU Ch. 7.8 - Prob. 2SWU Ch. 7.8 - Prob. 3SWU Ch. 7.8 - Prob. 4SWU Ch. 7.8 - Prob. 5SWU Ch. 7.8 - Prob. 6SWU Ch. 7.8 - Prob. 7SWU Ch. 7.8 - Prob. 8SWU Ch. 7.8 - Prob. 9SWU Ch. 7.8 - Prob. 10SWU Ch. 7.8 - Prob. 11SWU Ch. 7.8 - Prob. 12SWU Ch. 7.8 - Prob. 13SWU Ch. 7.8 - Prob. 14SWU Ch. 7.8 - Prob. 15SWU Ch. 7.8 - Prob. 16SWU Ch. 7.8 - Prob. 1E Ch. 7.8 - Finding Partial Integrals In Exercises 1-10, find...Ch. 7.8 - Prob. 3E Ch. 7.8 - Prob. 4E Ch. 7.8 - Prob. 5E Ch. 7.8 - Prob. 6E Ch. 7.8 - Prob. 7E Ch. 7.8 - Prob. 8E Ch. 7.8 - Prob. 9E Ch. 7.8 - Prob. 10E Ch. 7.8 - Prob. 11E Ch. 7.8 - Prob. 12E Ch. 7.8 - 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Prob. 19E Ch. 7.9 - Prob. 20E Ch. 7.9 - Prob. 21E Ch. 7.9 - Prob. 22E Ch. 7.9 - Prob. 23E Ch. 7.9 - Prob. 24E Ch. 7.9 - Prob. 25E Ch. 7.9 - Population Density The population density (in...Ch. 7.9 - Prob. 27E Ch. 7.9 - Prob. 28E Ch. 7.9 - Prob. 29E Ch. 7.9 - Prob. 30E Ch. 7.9 - Average Weekly Profit A firms weekly profit P (in...Ch. 7.9 - Prob. 32E Ch. 7.9 - Prob. 33E Ch. 7.9 - Prob. 34E Ch. 7.9 - Prob. 35E Ch. 7.9 - Prob. 36E Ch. 7 - Plotting Points in Space In Exercises 1 and 2,...Ch. 7 - Prob. 2RE Ch. 7 - Prob. 3RE Ch. 7 - Prob. 4RE Ch. 7 - Prob. 5RE Ch. 7 - Prob. 6RE Ch. 7 - Prob. 7RE Ch. 7 - Prob. 8RE Ch. 7 - Prob. 9RE Ch. 7 - Prob. 10RE Ch. 7 - Prob. 11RE Ch. 7 - Prob. 12RE Ch. 7 - Finding a Trace of a Sphere In Exercises 13 and...Ch. 7 - Prob. 14RE Ch. 7 - Prob. 15RE Ch. 7 - Prob. 16RE Ch. 7 - Prob. 17RE Ch. 7 - Prob. 18RE Ch. 7 - Prob. 19RE Ch. 7 - Prob. 20RE Ch. 7 - Prob. 21RE Ch. 7 - Prob. 22RE Ch. 7 - Prob. 23RE Ch. 7 - Prob. 24RE Ch. 7 - Prob. 25RE Ch. 7 - Prob. 26RE Ch. 7 - 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Prob. 80RE Ch. 7 - Prob. 81RE Ch. 7 - Prob. 82RE Ch. 7 - Prob. 83RE Ch. 7 - Prob. 84RE Ch. 7 - Prob. 85RE Ch. 7 - Prob. 86RE Ch. 7 - Evaluating a Double Integral In Exercises 85-88,...Ch. 7 - Prob. 88RE Ch. 7 - Prob. 89RE Ch. 7 - Prob. 90RE Ch. 7 - Finding Area with a Double Integral In Exercises...Ch. 7 - Finding Area with a Double Integral In Exercises...Ch. 7 - Finding the Volume of a Solid Region In Exercises...Ch. 7 - Prob. 94RE Ch. 7 - Prob. 95RE Ch. 7 - Prob. 96RE Ch. 7 - Prob. 97RE Ch. 7 - Prob. 98RE Ch. 7 - Prob. 99RE Ch. 7 - Prob. 100RE Ch. 7 - Prob. 101RE Ch. 7 - Average Revenue A company sells two products whose...Ch. 7 - Prob. 103RE Ch. 7 - Average Production Repeat Exercise 103 for the...Ch. 7 - Prob. 1TYS Ch. 7 - Prob. 2TYS Ch. 7 - Prob. 3TYS Ch. 7 - Prob. 4TYS Ch. 7 - Prob. 5TYS Ch. 7 - Prob. 6TYS Ch. 7 - Prob. 7TYS Ch. 7 - Prob. 8TYS Ch. 7 - Prob. 9TYS Ch. 7 - Prob. 10TYS Ch. 7 - Prob. 11TYS Ch. 7 - Prob. 12TYS Ch. 7 - Prob. 13TYS Ch. 7 - Prob. 14TYS Ch. 7 - Prob. 15TYS Ch. 7 - Prob. 16TYS Ch. 7 - Prob. 17TYS Ch. 7 - Prob. 18TYS Ch. 7 - Prob. 19TYS Ch. 7 - Prob. 20TYS Ch. 7 - Prob. 21TYS Ch. 7 - Prob. 22TYS

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The maximum value of the function f ( x , y ) = x 2 − y 2 subject to the constraint 2 y − x 2 = 0 by the use of Lagrange multipliers.

Solution Summary: The author explains how to calculate the maximum value of the function f(x,y) by the use of Lagrange multipliers.

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To calculate: The maximum value of the function f(x,y)=x2−y2 subject to the constraint 2y−x2=0 by the use of Lagrange multipliers.

Chapter 7 Solutions