Explanation Given: Thefunction to minimize f ( x , y ) = x 2 + y 2 − x y − 4 subject to x + y = 6 Concept Used: To optimize (that is, to find the minimum and maximum value of) a function f ( x , y ) subject to the constraint g ( x , y ) = k Solve the following system of equations ∇ f ( x , y ) = λ ∇ g ( x , y ) g ( x , y ) = k where λ is the Lagrange Multiplier. Plug all the solutions ( x , y ) from the above step into f ( x , y ) . The maximum value of the function f ( x , y ) subject to the constraint g ( x , y ) = k is the maximum value of f ( x , y ) after plugging all the solutions ( x , y ) into f ( x , y ) . Calculation: For the givenfunction f ( x , y ) = x 2 + y 2 − x y − 4 Calculating the first partial derivatives f x and f y of f ( x , y ) f ( x , y ) = x 2 + y 2 − x y − 4 that is, f x ( x , y ) = 2 x − y and, f y ( x , y ) = 2 y − x and, calculating the first partial derivatives g x and g y of g ( x , y ) g ( x , y ) = x + y that is, g x ( x , y ) = 1 and, g y ( x , y ) = 1 Substituting the values in the equation f x ( x , y ) = λ g x ( x , y ) that is, 2 x − y = λ ( 1 ) y = 2 x − λ - - - - - ( 1 ) and, the values in the equation f y ( x , y ) = λ g y ( x , y ) that is, 2 y −

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 11.8, Problem 9PS

To determine

To find:Minimize f(x,y)=x2+y2−xy−4 subject to constraint x+y=6 using the method of Lagrange multipliers.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.8, Problem 8PS

Chapter 11.8, Problem 10PS

Students have asked these similar questions

Find maximum and minimum value with langranfe multiplier

Suppose the labor cost (in dollars) for manufacturing a particular product can be approximated by 3 L(x, y) = 1⁄2 x² + y² − 5x − 6y − 2xy + 221 where x is the number of hours required by a skilled craftsperson and y is the number of hours required by a semiskilled person. What values of x and y will minimize cost and what is the minimum cost? Labor costs will be minimized when x = The minimum labor cost will be $ and y= =

2. Use the method of Lagrange multipliers to show that for x, y > 0, we have x"+y" > ()"

Chapter 11 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Minimize f ( x , y ) = x 2 + y 2 − x y − 4 subject to constraint x + y = 6 using the method of Lagrange multipliers.

Solution Summary: The author explains how to minimize f(x,y) using the method of Lagrange multipliers.

Videos

To find:Minimize f(x,y)=x2+y2−xy−4 subject to constraint x+y=6 using the method of Lagrange multipliers.

Want to see the full answer?

Chapter 11 Solutions