   Chapter 14.8, Problem 7E

Chapter
Section
Textbook Problem

Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.7. f(x, y, z) = 2x + 2y + 2z; x2 + y2 +z2 = 9

To determine

To find: The extreme value of the function f(x,y,z)=2x+2y+z subject to the constraint x2+y2+z2=9 by using Lagrange multipliers.

Explanation

Given:

The functionis f(x,y,z)=2x+2y+z subject to the constraint g(x,y)=x2+y2+z2=9 .

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx fy=λgy fz=λgz and g(x,y,z)=k ”.

Calculation:

The Lagrange multipliers f(x,y,z)=λg(x,y,z) is computed as follows,

f(x,y,z)=λg(x,y,z)fx,fy,fz=λgx,gy,gzfx(2x+2y+z),fy(2x+2y+z),fz(2x+2y+z)=λgx(x2+y2+z2),gy(x2+y2+z2),gz(x2+y2+z2)2,2,1=λ2x,2y,2z

Thus, the value of f(x,y,z)=λg(x,y,z) is 2,2,1=λ2x,2y,2z .

By the definition ey,xey=λ2x,2y it can be express as follows,

2=λ(2x) (1)

2=λ(2y) (2)

2=λ(2z) (3)

x2+y2+z2=9 (4)

From the equation (1),

2=λ(2x)x=1λ

The value of x=1λ .

From the equation (2),

2=λ(2y)y=1λ

Thus, the value of y=1λ .

From the equation (3),

1=λ(2z)z=12λ

The value of z=12λ .

Substitute x=1λ,y=1λandz=12λ in the equation (4) and obtain the value of λ as follows,

x2+y2+z2=9(1λ)2+(1λ)2+(12λ)2=94+4+14λ2=994λ2=9

Simplify further as,

94λ2=914λ2=14λ2=1λ=±12

Thus, the value of λ=±12

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