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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Positive Numbers In Exercises 33-36, fine three positive numbers x, y, and z that satisfy the giver conditions.

The product is 1 and the sum of the squares is a minimum

To determine

To calculate: The three positive numbers x, y and z if their product is 1 and the sum of the squares is a minimum.

Explanation

Given Information:

The product of three numbers x, y and z is 1 and the sum of the squares is a minimum.

Formula used:

If the function y=f(x) has a maximum at a point then their first partial derivatives fx(x) and fy(x) at that point are zero.

Calculation:

Consider the three numbers be x, y and z.

The product of three numbers x, y and z is 1.

xyz=1z=1xy

The sum S of the squares is a minimum.

S=x2+y2+z2

Substitute z=1xy in the equation S=x2+y2+z2,

S=x2+y2+(1xy)2

Now, the first partial derivatives are,

Sx=2x+21xy1y(1x2)=2(x1x3y2)

And

Sy=2y+21xy1x(1y2)=2(y1x2y3)

Put

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