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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 13-18, use Lagrange multipliers to find the indicated extremum. Assume that x, y, and z are positive. See Example 1.

Maximize  f ( x , y , z ) = x y z

Constraint : x   +   y   +   z   6   =   0

To determine

To calculate: The maximum value of the function f(x,y,z)=xyz subject to the constraint x+y+z6=0 by the use of Lagrange multipliers.

Explanation

Given Information:

The provided function is f(x,y,z)=xyz subject to the constraint x+y+z6=0. Consider that the variables x and y are positive.

Formula used:

Method of Lagrange multipliers,

If the function f(x,y,z) contains a maximum or minimum subject to the constraint g(x,y,z)=0 then the maximum or minimum can occur at one of the critical numbers of the function F is,

F(x,y,z,λ)=f(x,y,z)λg(x,y,z) where, λ is a Lagrange multiplier.

Steps to determine the minimum or maximum of the function f.

1. Solve the system of equations,

Fx(x,y,z,λ)=0Fy(x,y,z,λ)=0Fz(x,y,z,λ)=0Fλ(x,y,z,λ)=0

2. Determine the value of the function f at each solution obtained from the step 1.

The largest value gives the maximum value of function f subject to the constraint g(x,y,z)=0 and the lowest value gives the minimum value of function f subject to the constraint g(x,y,z)=0.

Calculation:

Consider the function, f(x,y,z)=xyz

The provided constraint is x+y+z6=0

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