Lagrange Multiplier MethodMaking Sense of the Results

[Saladsamurai]

Homework Statement

I am doing this lagrange multiplier problem with 2 constraints. I have completely solved it as shown in the image below. I have found that for lambda = 1 and mu = +/- 1/2 I have that x=+/- [sqrt(2)] y=+/- [1/sqrt(2)] and z=+/- [1/sqrt(2)].

So I am trying to figure out what points I actually have now. It seemed to me that since for x,y,z I have both a positive and negative value, I should have 2*2*2= 8 points to look at. But the solution only lists four. Am I messing this up somehow? Are there not 8 points given by the solution below? Thanks.

 

[Dick]

That's pretty good. But how many of those 8 points satisfy xy=1?

 

[Saladsamurai]

That's pretty good. But how many of those 8 points satisfy xy=1?

Ah ha. I see now. Thanks Dick! Is there a general approach to keeping track of which points are valid for all constraints? Or do you just solve the n equations for n unknowns and then back-check? I know there is probably no blanket rule.. but is that the approach more times than not?

 

[Dick]

Ah ha. I see now. Thanks Dick! Is there a general approach to keeping track of which points are valid for all constraints? Or do you just solve the n equations for n unknowns and then back-check? I know there is probably no blanket rule.. but is that the approach more times than not?

You've got it. No, I don't think there's any more general way than back checking. Your solutions from solving subsets of the equations may give you extraneous solutions. Just back check.

 

[Saladsamurai]

Thanks a bunch!

 

[cronxeh]

Well like Dick said you just got to check all the points to see which one is a max and which is min, but wolframalpha is there to help

http://www.wolframalpha.com/input/?i=Extrema+yz+xy,+xy=1,+y^2+z^2=1

 

[Saladsamurai]

Haha! Nice one cronxeh