Explanation Definition used: Lagrange Multipliers 1. The form of the constraint is g ( x , y ) = 0 . 2. The form of the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . 3 Form the system of equations F x ( x , y , λ ) = 0 , F y ( x , y , λ ) = 0 and F λ ( x , y , λ ) = 0 , compute the solution of these system and also the relative extrema among these solution. Calculation: Sum of three positive number is 90 and it’s product is maximum. That is, x + y + z = 90 and f ( x , y , z ) = x y z . The function f ( x , y , z ) = x y z is subject to the constrains x + y + z = 90 . The constrain in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = x + y + z − 90 The form of the Lagrange function is, F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . Substitute f ( x , y , z ) = x y z and g ( x , y ) = x + y + z − 90 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) , F ( x , y , z , λ ) = x y z − λ ( x + y + z − 90 ) = x y z − λ x − λ y − λ z + 90 λ Thus, the Lagrange function is F ( x , y , z , λ ) = x y z − λ x − λ y − λ z + 90 λ . Need to find the partial derivative of the function F ( x , y , z , λ ) . Partial derivative of the function F ( x , y , z , λ ) with respect to x and fix y , z and λ are constant. F x ( x , y , z , λ ) = ∂ ∂ x ( x y z − λ x − λ y − λ z + 90 λ ) = ∂ ∂ x ( x y z ) − ∂ ∂ x ( λ x ) − ∂ ∂ x ( λ y ) − ∂ ∂ x ( λ z ) + ∂ ∂ x ( 90 λ ) = ( y z ) ∂ ∂ x ( x ) − ( λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( λ y ) − ∂ ∂ x ( λ z ) + ∂ ∂ x ( 90 λ ) Simplify the terms and obtain the partial derivatives, F x ( x , y , z , λ ) = ( y z ) ( 1 ) − ( λ ) ( 1 ) − 0 − 0 + 0 = y z − λ Therefore, the partial derivative with respect to x is F x ( x , y , z , λ ) = y z − λ . The Partial derivative of the function F ( x , y , z , λ ) with respect to y and fix x , z and λ are constant. F y ( x , y , z , λ ) = ∂ ∂ y ( x y z − λ x − λ y − λ z + 90 λ ) = ∂ ∂ y ( x y z ) − ∂ ∂ y ( λ x ) − ∂ ∂ y ( λ y ) − ∂ ∂ y ( λ z ) + ∂ ∂ y ( 90 λ ) = ( x z ) ∂ ∂ y ( y ) − ∂ ∂ y ( λ x ) − ( λ ) ∂ ∂ y ( y ) − ∂ ∂ y ( λ z ) + ∂ ∂ y ( 90 λ ) = ( x z ) ( 1 y 1 − 1 ) − 0 − ( λ ) ( 1 y 1 − 1 ) − 0 + 0 Simplify the terms and obtain the partial derivatives, F y ( x , y , z , λ ) = ( x z ) ( 1 ) − ( λ ) ( 1 ) = x z − λ Therefore, the partial derivative with respect to x is F y ( x , y , z , λ ) = x z − λ

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 17.4, Problem 13E

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To find: Three positive numbers whose sum is 90 and whose product is a maximum.

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Chapter 17.4, Problem 12E

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Finite Mathematics and Calculus with Applications

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Three positive numbers whose sum is 90 and whose product is a maximum.

Solution Summary: The author explains how to find three positive numbers whose sum is 90 and their product is a maximum. The form of the Lagrange function is F(x,y,lambda

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Chapter 17 Solutions