Find extreme values for function of 3 variables and two constraints

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SUMMARY

The discussion centers on solving a problem involving finding extreme values for a function of three variables under two constraints using Lagrange multipliers. The user successfully computed the gradients of the function and constraints, denoted as fgrad, ggrad, and hgrad, and set up the equation fgrad = λ*ggrad + μ*hgrad. The user seeks clarification on determining the minimum and maximum values of the function and requests verification of their algebraic calculations for the variables x, y, z, λ, and μ.

PREREQUISITES
  • Understanding of Lagrange multipliers
  • Familiarity with gradient vectors
  • Knowledge of constrained optimization
  • Basic algebra skills for solving equations
NEXT STEPS
  • Study the method of Lagrange multipliers in detail
  • Learn how to identify and classify extreme values in multivariable calculus
  • Practice solving optimization problems with multiple constraints
  • Review algebraic techniques for solving systems of equations
USEFUL FOR

Students in calculus or optimization courses, mathematicians, and anyone interested in advanced problem-solving techniques in multivariable functions.

jimbo71
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Homework Statement


see problem 7 attachement


Homework Equations





The Attempt at a Solution


see problem 7 attempt attachment
I found the fgrad and ggrad and hgrad. I set fgrad equal to lambda*ggrad+mu*hgrad. using the two constraint equations I was able to solve for x,y,z,lambda,mu. However again I am not sure what to do with these values. Also in this problem is states find the extreme values. does that mean I need to find MIN and MAX. and if so how do i do that. Also can someone please check to see if my algebra is correct in solving for x,y,z,lamdba,mu. Thank you
 

Attachments

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  • 7 attempt.jpg
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