Explanation Given: The length of the diagonal of the rectangular box is L . Definition used: “The Lagrange multipliers defined as ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) . This equation can be expressed as f x = λ g x , f y = λ g y , f z = λ g z and g ( x , y , z ) = k ”. Calculation: Let the dimensions be x , y and z . Then, the volume of the rectangular box is V = f ( x , y , z ) = x y z where x > 0 , y > 0 , z > 0 The length of the diagonal of the rectangular box is g ( x , y , z ) = L = x 2 + y 2 + z 2 . Thus, the maximum value of the function f ( x , y , z ) = x y z subject to the constraint g ( x , y , z ) = x 2 + y 2 + z 2 = L 2 . The Lagrange multipliers ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) is computed as follows, ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) 〈 f x , f y , f z 〉 = λ 〈 g x , g y , g z 〉 〈 f x ( x y z ) , f y ( x y z ) , f z ( x y z ) 〉 = λ 〈 g x ( x 2 + y 2 + z 2 ) , g y ( x 2 + y 2 + z 2 ) , g z ( x 2 + y 2 + z 2 ) 〉 〈 y z , x z , x y 〉 = λ 〈 2 x , 2 y , 2 z 〉 Thus, the value of ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) is 〈 y z , x z , x y 〉 = λ 〈 2 x , 2 y , 2 z 〉

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

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ISBN: 9781305779198

Author: James Stewart

Publisher: Cengage Learning

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Chapter 14.8, Problem 43E

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Chapter 14.8, Problem 42E

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Chapter 14 Solutions

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

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The largest volume of the rectangular box by using Lagrange multipliers.

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Chapter 14 Solutions