MIN and MAX for funtions of two and three variables with constrainments

In summary, the conversation is about solving equations for minimum and maximum values using gradient vectors and constraint equations. The speaker is asking for help with problem 4 and also mentions a similar problem 5. They also ask for confirmation on their algebra and inquire about the relationship between the variables and the minimum and maximum values.
  • #1
jimbo71
81
0

Homework Statement


see problem 4 and 5 attachment


Homework Equations





The Attempt at a Solution


see problem 4 attachement
I found the gradient vectors of each and set fgrad=lamdbda*ggrad and used the constrainment equation to solve for all three variables. What is confusing me is I'm not sure what to do with the x,y,lambda values. How do they relate to the minimum and maximum values? I cannot attachemt my problem 5 attempt because it is too large of a file. but they are essential the same type of problem with different equations so if someone could please help me on number four i'll probably figure out number five. Also, can you check my that my algebra is correct in solving x,y,lambda for problem 4. Thank you!
 

Attachments

  • problem 4 and 5.jpg
    problem 4 and 5.jpg
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  • 4 attempt.jpg
    4 attempt.jpg
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  • #2
i need help asap
 
  • #3
first, if 2xy^4=L*2x then either case#1 x=0 or case#2 x does not =0. if not, then you can solve L=y^4.
put this into the next equation to eliminate L. check both cases.

second, if 4x^2y^3=L*4y then AGAIN, case#3 y=0 or case#4 y does not =0.

eliminate L and put all these cases into the last equation which will probably give multiple solutions for each case. once you have a point x=? y=? put this into F(x,y)=? which one is smallest/largest?
 

1. What is the purpose of using MIN and MAX in functions of two and three variables with constraints?

The purpose of using MIN and MAX in functions with constraints is to determine the minimum and maximum values of the function within a specified range of values for the variables. This can help in optimizing a function and finding the optimal solution.

2. How do you find the minimum and maximum values of a function with constraints?

To find the minimum and maximum values of a function with constraints, you can use the method of Lagrange multipliers. This involves finding the gradient of the function and the constraint, setting them equal to each other, and solving for the variables.

3. Can MIN and MAX be used in any type of function?

Yes, MIN and MAX can be used in any type of function, including linear, quadratic, and exponential functions. They are commonly used in optimization problems to find the best possible solution within a given set of constraints.

4. Are there any limitations to using MIN and MAX in functions of two and three variables?

One limitation is that the method of Lagrange multipliers may not always produce the correct solution. Additionally, the number of variables and constraints can greatly impact the complexity of the problem and the time needed to find the optimal solution.

5. How can MIN and MAX be applied in real-world scenarios?

MIN and MAX can be applied in numerous real-world scenarios such as resource allocation, production optimization, and portfolio management. They can also be used in engineering and scientific research to find the best design or solution for a given set of constraints.

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