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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

(a) Find the maximum value of

f ( x 1 , x 2 , . . . , x n ) = x 1 x 2 . . . x n n

f ( x 1 , x 2 , . . . , x n ) = x 1 x 2 . . . x n n

given that x1, x2, . . . , xn are positive numbers and x1 + x2 + . . . + xn = c, where c is a constant.

(b) Deduce from part (a) that if x1, x2, . . . , xn are positive numbers, then

x 1 x 2 . . . x n n x 1 + x 2 + . . . + x n n

This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?

(a)

To determine

To find: The maximum value of f(x1,x2,...,xn)=x1x2...xnn .

Explanation

Given:

The given function is f(x1,x2,...,xn)=x1x2...xnn where x1,x2,...,xn are positive numbers and x1+x2+...+xn=c , c is constant.

Calculation:

The given function is f(x1,x2,...,xn)=x1x2...xnn .

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(x1x2...xnn),fx2(x1x2...xnn),fxn(x1x2...xnn)=λgx1(x1+x2+...+xnc),gx2(x1+x2+...+xnc),gxn(x1+x2+...+xnc)[1n(x1x2...xn)1n1(x2x3...xn)],[1n(x1x2...xn)1n1(x1x3...xn)]...[1n(x1x2...xn)1n1(x1x2...xn1)]=λ1,1,...,1

Thus, the value of f(x1,x2,...,xn)=λg(x1,x2,...,xn) is {[1n(x1x2...xn)1n1(x2x3...xn)],[1n(x1x2...xn)1n1(x1x3...xn)]...[1n(x1x2...xn)1n1(x1x2...xn1)]}=λ1,1,...,1

The result,

{[1n(x1x2...xn)1n1(x2x3...xn)],[1n(x1x2...xn)1n1(x1x3...xn)]...[1n(x1x2

(b)

To determine

To show: Thevalues x1,x2,...,xn are positive numbers then prove x1x2...xnnx1+x2+...+xnn is equal under which situations.

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Chapter 14 Solutions

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