Lagrange Multiplier where constraint is a rectangle

[SamitC]

Hello,
How can I use Lagrange Multipliers to get the Extrema of a curve f(x,y) = x²+4y²-2x²y+4 over a rectangular region -1<=x<=1 and -1<=y<=1 ?
Thanks

 

[Ssnow]

yes, before it is simpler to study the critical point of the gradient and look for max or min inside the rectangle, you can apply the Lagrange method for the boundary of the rectangle ...

 

[stevendaryl]

Just to add to what Ssnow said, I think that Lagrange multipliers are only directly useful for continuous constraints. If the constraint is just that $(x,y)$ must be inside a rectangle, then I would think that you would do the following:

1. First, find the extrema forgetting about the rectangle.
2. Then if the extrema found in step 1 are all inside the rectangle, then you're done.
3. If not, then use the method of Lagrange multipliers to find the extrema for each of the four sides of the rectangle.

 

[Orodruin]

I actually think it is overkill to use Lagrange multipliers for the boundaries. Since the boundaries are so simple, just insert the boundary constraint and treat the resulting function as a function of the remaining variable.

 

[SamitC]

Thanks a lot. Its clear now.