Explanation Given Information: The function f ( x , y ) = x y subject to the constraint x − y = 6 . Formula Used: The extreme value of the function f ( x , y ) subject to the constraint g ( x , y ) can be obtained in the following steps: 1. Form a Lagrange function F ( x , y , λ ) = f ( x , y ) − λ g ( x , y ) . 2. Obtain the solutions for the system F x = 0 , F y = 0 , F λ = 0 as ( x , y , λ ) . 3. Then check whether the solution ( x , y ) is a maximum or a minimum value using test values from the constraints. Calculation: Consider the function, f ( x , y ) = x y Rewrite the provided constraint as: x − y = 6 x − y − 6 = 0 Now, form a new function F given by, F ( x , y , λ ) = x y − λ ( x − y − 6 ) Find F x , F λ and F y . For F x , x is treated as variable z , y and λ are treated as constants and the equation is differentiated. F x = y − λ ….. ( 1 ) For F y , y is treated as variable and x , z and λ are treated as constants and the equation is differentiated. F y = x + λ ….. ( 2 ) For F , λ is treated as variable and x and y are treated as constants and the equation is differentiated

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Author: Marvin Bittinger, David Ellenbogen

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Chapter 6.5, Problem 16E

To determine

To calculate: The two numbers that have minimum product among numbers with difference 6.

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Chapter 6.5, Problem 15E

Chapter 6.5, Problem 17E

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