Find lagrangian (please check work)

[939]

Homework Statement

F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

Form the lagrangian.

Homework Equations

F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

The Attempt at a Solution

My only question is solving 11 = x + y

My book says the answer is:
L = 96xy - 4x + λ(11 - x - y)

But I got:
L = 96xy - 4x + λ(x + y - 11)

Just to confirm, both are correct because it just depends how you solve the constraint, no?

 

[Ray Vickson]

Homework Statement

F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

Form the lagrangian.

Homework Equations

F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

The Attempt at a Solution

My only question is solving 11 = x + y

My book says the answer is:
L = 96xy - 4x + λ(11 - x - y)

But I got:
L = 96xy - 4x + λ(x + y - 11)

Just to confirm, both are correct because it just depends how you solve the constraint, no?

It makes no difference: one λ will just have the opposite sign of the other.

However, if does matter when you are doing post-optimality analysis. For example, you can use the value of λ to find the approximate change in the optimal value of F when the constraint changes to x+y = 11.1, for example. In that case you need to understand exactly which form of Lagrangian to use, or at least, how to apply either +λ or -λ in the analysis. It also matters whether you are maximizing or minimizing, and in your post you did not say which you were doing.

 

[939]

It makes no difference: one λ will just have the opposite sign of the other.

However, if does matter when you are doing post-optimality analysis. For example, you can use the value of λ to find the approximate change in the optimal value of F when the constraint changes to x+y = 11.1, for example. In that case you need to understand exactly which form of Lagrangian to use, or at least, how to apply either +λ or -λ in the analysis. It also matters whether you are maximizing or minimizing, and in your post you did not say which you were doing.

The goal was to find an optimal value subject to the constraint... Does it matter then, and how do you know which one to pick?

Also, both partial derivatives would be correct, right?

 

[Ray Vickson]

The goal was to find an optimal value subject to the constraint... Does it matter then, and how do you know which one to pick?

Also, both partial derivatives would be correct, right?

What is optimal? Maximum? Minimum?

I don't know what you mean when you ask if both partials are correct; you did not give formulas for the partials, so I have no way to know if they are correct or not.

 

[939]

What is optimal? Maximum? Minimum?

I don't know what you mean when you ask if both partials are correct; you did not give formulas for the partials, so I have no way to know if they are correct or not.

Sorry.

I mean if you were NOT asked to find a maximum or minimum, and were ONLY asked to find the lagrangian and the three partial derivatives - would these two answers be correct regardless of how you solved the constraint function?

 

[Ray Vickson]

Sorry.

I mean if you were NOT asked to find a maximum or minimum, and were ONLY asked to find the lagrangian and the three partial derivatives - would these two answers be correct regardless of how you solved the constraint function?

No: you need to solve the equations correctly! I think maybe you meant to say "regardless of which form of Lagrangian you use". Then the answer would be yes, if you make no errors during solving.

However: you don't really need to ask; you can just go ahead and do it both ways to see what you get. In fact, that would be faster than submitting a question and waiting for an answer!