How Do You Solve Lagrange Multipliers with Complex Constraints?

[kawsar]

1. Use the method of Lagrange multipliers to nd the minimum value of
the function:
f(x,y,z) = xy + 2xz + 2yz

subject to the constraint xyz = 32.



I understand the method how Lagranges Multipliers is donw done but seem to have got stuck solving the Simultaneous Equations involving the Partial Derivatives involving \lambda.

I think the 3 Partial Derivatives (set equal to 0) are:

f_{x}=y+2z-\lambdayz=0
f_{y}=x+2z-\lambdaxz=0
f_{z}=2x+2y-\lambdaxy=0

Any chance helping me work out how I can solve x, y and z in terms of \lambda OR if I've made an earlier mistake somewhere, sorting that out for me?

Thanks

edit: f_{x} is supposed to be f sub x - Don't know how to write that with the editor.

 

[hotvette]

Your forgot one equation (the constraint itself). This gives you 4 equations in 4 unknowns.

 

[kawsar]

Ok. You mean the constraint itself and not the partial derivative in respect to Lambda.

I'll see if that gets me anywhere. Thanks!

 

[kawsar]

Sorry guys... A few more pointers please! Don't know what's wrong with me but can't seem to solve the eqns for some reason...

 

[hotvette]

Here is how I solved it:

1. Combine 1st and 2nd equations to eliminate the term containing Lambda. Result is an expression relating x & y

2. Combine 1st and 3rd equations to eliminate the term containing Lambda. Result is an expression relating x and z

3. Use the above and the constraint equation and to solve for z

4. Use the value of z to find x & y

 

[kawsar]

Thanks! When you say combine do you mean write one equals the other and then try and simplify it? Could you do your step one please.

 

[hotvette]

Thanks! When you say combine do you mean write one equals the other and then try and simplify it? Could you do your step one please.

The concept is basic Gauss elimination. Let's say you have the following:

2x + 3y = 5
x + 2y = 3

to eliminate the term with y, multiple the 1st equation by 2 and 2nd by 3:

4x + 6y = 10
3x + 6y = 9

Subtracting the 2nd from the 1st (i.e. combine) you get:

x = 1

Now apply the same idea to your situation to eliminate the term with Lambda.

 

[kawsar]

Hmmm... The first two formulae have 3 different variables and the Lambda. Will the basic Simultaneous Equation solving method work with that?

 

[hotvette]

Hmmm... The first two formulae have 3 different variables and the Lambda. Will the basic Simultaneous Equation solving method work with that?

Worked for me. Try it.