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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 19–22, find three positive numbers x, y, and z that satisfy the given conditions.

The sum is 90 and the product is a maximum.

To determine

To calculate: The values of x,y,z such that the product is maximum and the sum of the three numbers is 90.

Explanation

Given Information:

The x,y,z are three numbers such that the product is maximum and the sum of the three numbers is 90.

Formula used:

If f(x,y,z) has a maximum or minimum subject to the constraint g(x,y,z)=0, then it will

occur at one of the critical points of the function F defined by:

F=f(x,y,z)λg(x,y,z)

The variable λ is called the Lagrange Multiplier. To find the maximum or minimum of f follow

the given steps:

Step 1: Solve the following system of equations:

Fx(x,y,z,λ)=0Fy(x,y,z,λ)=0Fz(x,y,z,λ)=0Fλ(x,y,z,λ)=0

Step 2: Evaluate f at each solution point obtained in the first step. The greatest value yields maximum of f and the smallest value yields minimum of f subject to the constraint g(x,y,z)=0.

Calculation:

Consider the given equations,

f(x,y,z)=xyz

g(x,y,z)=x+y+z90=0

Now, consider the primary equation,

F=f(x,y,z)λg(x,y,z)F=xyzλ(x+y+z90)

For critical numbers of F, differentiate F with respect to x, y,z. then set it equal to zero.

Fx=0Fx(x,y,z,λ)=0yzλ=0

And,

Fy=0Fy(x,y,z,λ)=0xzλ=0

And,

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