   Chapter 14, Problem 61RE

Chapter
Section
Textbook Problem

Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s).61. f(x, y, z) = xyz; x2 + y2 + z2 = 3

To determine

To find: The extreme values of the function f(x,y,z)=xyz subject to the constraint x2+y2+z2=3 by using Lagrange multipliers.

Explanation

Given:

The function f(x,y,z)=xyz which is subject to the constraint g(x,y,z)=x2+y2+z2=3 .

Result 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)=λg(x,y) is computed as follows,

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

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

The result yz,xz,xy=λ2x,2y,2z , can be expressed as follows.

yz=λ(2x) (1)

xz=λ(2y) (2)

xy=λ(2z) (3)

The extreme values of the function f(x,y,z)=xyz is computed as follows.

From the equations (1), (2) and (3),

λ=yz2x=xz2y=xy2z

Consider yz2x=xz2y and obtain y ,

yz2x=xz2yy2z=x2zy2=x2

Consider xz2y=xy2z and obtain z ,

xz2y=xy2zxz2=xy2z2=y2

Substitute y2=x2 and z2=y2 in the equation g(x,y,z)=x2+y2+z2=3 and obtain x,yandz ,

x2+y2+z2=3x2+x2+x2=33x2=3x=±1

Thus, the points are (1,1,±1),(1,1,±1),(1,1,±1)and(1,1,±1)

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