Hi JacobOCD have you tried using the ''jacobian''...

Sorry I realy tough it was funny...

I kind of feel bad now...

I'll try to help you then!

I think the problem is your ''Lagrangean'' (Well I for sure haven't seen it that way, I can't say it's wrong, but I've no idea what that is)

The method for Lagrange multiplicator I've in my school book say something like that (It's in french so I'll translate)

To find the min and max value of f(x,y,z) under a cibstraubt function g(x,y,z)=k (There are some stuff you have to make sure, like the derivate exist.)

find every value of x,y,z so that:

then evalaute f with those point

Note

is the same as

Now this is for the student book I've.

This is how I leanred lagrange multiplcator.

Does it help a bit more?

note: You said: "and I proceed taking partial derivatives, putting them equal to zero and thus creating a system of equations:"

Tacking the partial derivates is a step in lagrange multiplacator altough you never put them equal to zero. the equation you've to solve

If that wasn't clear enough let me know I'll try to explain more.