Explanation Result used: Method of Lagrange Multipliers “The below four steps are the procedure for method of Lagrange Multipliers: 1. Rewrite the problem in the form of maximize (or minimize) the function z = f ( x , y ) subject to the constraint g ( x , y ) = 0 . 2. Form the function F : F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . 3. Compute the critical points of F ; that is, the solution of the system of the equation F x ( x , y , z ) = 0 F y ( x , y , z ) = 0 F z ( x , y , z ) = 0 4. If ( x 0 , y 0 , λ 0 ) is the only critical point of the function F , then the point ( x 0 , y 0 ) is the solution of the given hypothesis. If F has more than one critical point, compute the value of f ( x , y ) at ( x 0 , y 0 ) for each critical point ( x 0 , y 0 , λ 0 ) of F . then assume that the largest value of these values is the maximum value of f ( x , y ) , subject to constraint g ( x , y ) = 0 , and the smallest is the minimum of f ( x , y ) , subject to the constraint g ( x , y ) = 0 .” Calculation: Step 1: Maximize f ( x , y ) = y + x y 2 Subject to: g ( x , y ) = x + y 2 − 1 = 0 Step 2: Form the function F by substituting f ( x , y ) = y + x y 2 and g ( x , y ) = x + y 2 − 1 in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) , F ( x , y , λ ) = y + x y 2 + λ ( x + y 2 − 1 ) = y + x y 2 + λ x + λ y 2 − λ Thus, the function F ( x , y , λ ) = y + x y 2 + λ x + λ y 2 − λ . Step 3: In order to find the critical points, first find the partial derivatives. Compute the partial derivative F x ( x , y , λ ) . That is, differentiate the function F ( x , y , λ ) with respect to x by taking y and λ are constants. F x ( x , y , λ ) = ∂ ∂ x ( y + x y 2 + λ x + λ y 2 − λ ) = ∂ ∂ x ( y ) + ∂ ∂ x ( x y 2 ) + ∂ ∂ x ( λ x ) + ∂ ∂ x ( λ y 2 ) − ∂ ∂ x ( λ ) = 0 + ( y 2 ) ∂ ∂ x ( x ) + ( λ ) ∂ ∂ x ( x ) + 0 − 0 = y 2 + λ Therefore, the partial derivatives with respect to x is F x ( x , y , λ ) = y 2 + λ . Compute the partial derivative F y ( x , y , λ ) . That is, differentiate the function F ( x , y , λ ) with respect to y by taking x and λ are constants

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Publisher: PEARSON

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Chapter 7.4, Problem 21E

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To find: The maximum value of the function f(x,y)=y+xy2 subject to constraint x+y2=1.

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Chapter 7.4, Problem 20E

Chapter 7.4, Problem 22E

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Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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The maximum value of the function f ( x , y ) = y + x y 2 subject to constraint x + y 2 = 1 .

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To find: The maximum value of the function f(x,y)=y+xy2 subject to constraint x+y2=1.

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Chapter 7 Solutions