Find extreme values for function of 3 variables and two constraints

In summary, the conversation involves a problem with an attachment and the attempt at solving it. The individual was able to find the fgrad, ggrad, and hgrad and set fgrad equal to lambda times ggrad plus mu times hgrad. They used two constraint equations to solve for x, y, z, lambda, and mu. However, they are unsure what to do with these values and are also unsure if they need to find minimum and maximum values. They also requested someone to check their algebra and needed help urgently.
  • #1
jimbo71
81
0

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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  • #2
i need help asap
 

1. How do you define extreme values for a function of 3 variables and two constraints?

The extreme values for a function of 3 variables and two constraints are the maximum and minimum values that the function can attain while satisfying the given constraints.

2. What are the steps to find extreme values for a function of 3 variables and two constraints?

To find extreme values for a function of 3 variables and two constraints, you can follow these steps:

  • Use the given constraints to eliminate one variable from the function.
  • Find the critical points by setting the partial derivatives of the function equal to zero.
  • Check the critical points to see if they satisfy the given constraints.
  • Evaluate the function at the critical points and the endpoints of the constraint region.
  • The highest and lowest values obtained will be the maximum and minimum extreme values, respectively.

3. Can the extreme values of a function of 3 variables and two constraints be found analytically?

Yes, the extreme values of a function of 3 variables and two constraints can be found analytically by using the above-mentioned steps. However, for complex functions, it may be easier to use numerical methods to find the extreme values.

4. How do the constraints affect the extreme values of a function of 3 variables?

The constraints determine the region in which the function is being evaluated. This means that the extreme values of the function will be limited to this region. Also, the constraints may restrict the possible values of the variables, which can affect the critical points and the endpoints of the constraint region.

5. Is it possible to have multiple extreme values for a function of 3 variables and two constraints?

Yes, it is possible to have multiple extreme values for a function of 3 variables and two constraints. These values can be either maximum or minimum, and they can occur at different points within the constraint region. It is important to check all critical points and endpoints to identify all possible extreme values.

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