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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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Using Lagrange Multipliers In Exercises 13–18, use Lagrange multipliers to find the indicated extremum. Assume that x, y, and z are positive. See Example 1.

Maximize f(x, y, z) = x + y + z

Constraint: x2 + y2 + z2 = 1

To determine

To calculate: The extremum of the function f(x,y,z)=x+y+z subjected to the constraint

g(x,y,z)=x2+y2+z21=0.

Explanation

Given Information:

The function f(x,y,z)=x+y+z subject to the constraint g(x,y,z)=x2+y2+z21 and x,y,z are positive numbers.

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)=x+y+z

g(x,y,z)=x2+y2+z21=0

Now, consider the primary equation,

F(x,y,z,λ)=f(x,y,z)λg(x,y,z)=x+y+zλ(x2+y2+z21)

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)=012λx=0

And,

Fy=0Fy(x,y,z,λ)=012λy=0

And,

Fz=0

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