   Chapter 7.6, Problem 16E ### 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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# 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.Minimize f ( x , y , z ) =   2 x 2 +   3 y 2 + 2 z 2 Constraint: x   +   y   +   z   −   24   =   0

To determine

To calculate: The extremum of the function f(x,y,z)=2x2+3y2+2z2 subjected to the constraint g(x,y,z)=x+y+z24=0.

Explanation

Given Information:

f(x,y,z)=2x2+3y2+2z2 subject to the constraint g(x,y,z)=x+y+z24 and x, y, z are

positive numbers.

Formula used:

If f(x,y,z) has a maximum or minimum subject to the constraint g(x,y,z)=0, then it will

get at one of the critical points for the function F defined as:

F=f(x,y,z)λg(x,y,z)

The variable λ is called the Lagrange Multiplier. To get the maximum or minimum of f follow

the given steps below:

Step1: Solve the following system of equations:

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

Step2: Calculate f at every solution point get in the first step. The highest value yields maximum of f and the smallest value yields minimum of f subject to the constraint g(x,y,z)=0.

Calculation:

Consider the given equations

f(x,y,z)=2x2+3y2+2z2g(x,y,z)=x+y+z24=0

Now, consider the primary equation

F=f(x,y,z)λg(x,y,z)F=2x2+3y2+2z2λ(x+y+z24)

For critical numbers of F, differentiate F with respect to x,y,z. then set it equal to zero

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