   Chapter 14.7, Problem 46E

Chapter
Section
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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