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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using Lagrange Multipliers In Exercises 73-78, use Lagrange multipliers to find the indicated extremum. Assume that the variables are positive

Maximize f ( x ,   y ) = 2 x y

Constraint: 2 x + y   12   =   0

To determine

To calculate: The maximum value of the function f(x,y)=2xy subject to constrain 2x+y12=0 by the application of Lagrange multipliers.

Explanation

Given Information:

The provided function is f(x,y)=2xy and the subject to constraints are 2x+y12=0.

Formula used:

To find the maximum or minimum values of function f follow the steps as,

Step 1: Solve the provided system of equations as;

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Step 2: Next calculate the function f at each of the solution point obtained from first step where the greatest value obtained gives the maximum of function f subject to constraint g(x,y)=0 and the least value gives the minimum of function f subject to constraint g(x,y)=0.

Calculation:

First let f(x,y,z)=2xy and g(x,y,z)=2x+y12=0.

Then define a new function F as,

F(x,y,z,λ)=f(x,y,z)λg(x,y,z)=2xyλ(2x+y12)

Now solve the equation by finding the partial derivatives of F with respect to x, y, z and λ as,

Fx(x,y,λ)=2y2λFy(x,y,λ)

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