   Chapter 14.8, Problem 14E

Chapter
Section
Textbook Problem

Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.14. f ( x 1 ,     x 2 ,     .     .     .     ,     x n )   =   x 1     +     x 2   +     .     .     .     +   x n ;     x 1 2   +   x 2 2   +     .     .     .     +     x n 2   =   1

To determine

To find: The extreme values of the function f(x1,x2,,xn)=x1+x2++xn subject to the constraint x12+x22++xn2=1 by using Lagrange multipliers.

Explanation

Given:

The function is f(x1,x2,,xn)=x1+x2++xn subject to the constrain x12+x22++xn2=1 .

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx fy=λgy fz=λgz and g(x,y,z)=k ”.

Calculation:

The Lagrange multipliers f(x1,x2,...,xn)=λg(x1,x2,...,xn) is computed as follows,

f(x1,x2,,xn)=λg(x1,x2,,xn)fx1,fx2,...,fxn=λgx1,gx2,...,gxnfx1(x1+x2++xn),fx2(x1+x2++xn),...fn(x1+x2++xn)=λgx1(x12+x22++xn2),gx2(x12+x22++xn2),...gxn(x12+x22++xn2)1,1,...,1=λ2x1,2x2,...,2xn

Thus, the value of f(x1,x2,...,xn)=λg(x1,x2,...,xn) is 1,1,...,1=λ2x1,2x2,...,2xn .

Frrom the result 1,1,...,1=λ2x1,2x2,...,2xn , the extreme values of the function f(x1,x2,,xn)=x1+x2++xn is computed as follows,

Equate the values of λ=12x1,λ=12x2,

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