4. The function has a maximum and minimum extreme value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x,y) = x² + y² + z², xyz = 4

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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4.
The function has a maximum and minimum extreme value. Use Lagrange multipliers to find the extreme values of the function
subject to the given constraint.
f(x,y) = x² + y² + z², xyz = 4
Transcribed Image Text:4. The function has a maximum and minimum extreme value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x,y) = x² + y² + z², xyz = 4
Expert Solution
Step 1

Given fx,y,z=x2+y2+z2 and constraint xyz=4

Let gx,y,z=xyz-4

fx=λgxfy=λgyfz=λgz

where λ is the lagrange multiplier.

2x=λyz           . . . (1)2y=λxz           . . . (2)2z=λxy           . . . (3)

Step 2

Solving by multiplying both sides of the equation (1), (2) and (3) by x, y and z respectively,

2x2=λxyz2y2=λxyz2z2=λxyz

Equating,

2x2=2y2=2z2x2=y2=z2x=y=z

Substituting back to the constrained equation,

xxx=4x3=4x=43  =1.58=y=z

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