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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

To determine

To find: The three positive numbers whose sum is 12 and the sum of whose squares is small.

Explanation

Given:

The sum of three numbers is 12.

The square of the numbers is small.

Calculation:

Let the three numbers be x, y, z.

Then the equation is, x+y+z=12 and f(x,y)=x2+y2+z2 .

Rewrite the equation as,

x+y+z=12z=12xy

Substitute z=12xy in the function f(x,y)=x2+y2+z2 .

Thus, f(x,y)=x2+y2+(12xy)2 .

Take the partial derivative of f(x,y) with respect x and obtain fx(x,y) .

fx=x(x2+y2+(12xy)2)=x(x2)+x(y2)+x(12xy)2=2x+0+2(12xy)(1)=2x2(12xy)

Simplify further as follows.

fx=2x24+2x+2y=4x+2y24

Hence, fx=4x+2y24 (1)

Take the partial derivative of f(x,y) with respect y and obtain fy(x,y) .

fy=y(x2+y2+(12xy)2)=y(x2)+y(y2)+y(12xy)2=0+2y+2(12xy)(1)=2y2(12xy)

Simplify further as follows.

fy=2y24+2x+2y=2x+4y24

Thus, fy=2x+4y24 (2)

Set the above partial derivatives to 0 and find the values of x and y.

From the equation (1),

4x+2y24=04x+2y=242y=244xy=122x

Substitute y=122x in the equation (2),

2x+4(122x)24=02x+488x24=06x=24x=4

Substitute x=4 in the equation y=122x

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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