Explanation Given: Thefunction to minimize f ( x , y ) = x 2 + y 2 subject to x + y = 24 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 Calculating the first partial derivatives f x and f y of f ( x , y ) f ( x , y ) = x 2 + y 2 that is, f x ( x , y ) = 2 x and, f y ( x , y ) = 2 y 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

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 3PS

To determine

To find: Minimize f(x,y)=x2+y2 subject to constraint x+y=24 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 2PS

Chapter 11.8, Problem 4PS

Students have asked these similar questions

5. Use the Lagrange multiplier method to find the values of x and that minimise the function /2x2 + y? subject to the constraint x +y = 1.

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 Lagrange multipliers to maximize f(x, y) 2.x + y = 1. = x + 3xy + 4y subject to the constraint

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 subject to constraint x + y = 24 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 subject to constraint x+y=24 using the method of Lagrange multipliers.

Want to see the full answer?

Chapter 11 Solutions