   Chapter 7.6, Problem 1CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Find the maximum volume of V   =   x y z subject to the constraint 2 x   + 4 y   + z −   8   =   0.

To determine

To calculate: The volume of function V=xyz subject to the constraint 2x+4y+z8=0 by the use of Lagrange multipliers.

Explanation

Given Information:

The provided function for the volume is V=xyz subject to the constraint 2x+4y+z8=0. Consider that the variables x and y are positive.

Formula used:

Method of Lagrange multipliers,

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

F(x,y,λ)=f(x,y)λg(x,y) 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,λ)=0Fy(x,y,λ)=0Fλ(x,y,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)=0 and the lowest value gives the minimum value of function f subject to the constraint g(x,y)=0.

Calculation:

Consider the function,

f(x,y,z)=V=xyz

The provided constraint is 2x+4y+z8=0.

So, g(x,y)=2x+4y+z8

Now, the new function F is,

F(x,y,λ)=xyzλ(2x+4y+z8)

Find the partial derivative of function F to determine the critical number for F.

The partial derivatives of F with respect to x and then set it to zero.

Fx(x,y,λ)=x[xyzλ(2x+4y+z8)]0=yz2λλ=yz2

The partial derivatives of F with respect to y and then set it to zero.

Fy(x,y,λ)=y[xyzλ(2x+4y+z8)]0=xz4λ

The partial derivatives of F with respect to z and then set it to zero

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