Hello. I am trying to teach myself the Lagrangean method for purposes of constrained optimization, and I've had a couple of shots at a problem which I, uh, haven't been able to solve for some reason. Mostly because it degenerates to algebraic nonsense, I think. I would appreciate if someone took a look at the first part of my solution, so I can be sure that I am not messing it up before I start solving the equation system it yields, which obviously can be messed up easily.

I have an objective function (which is to be minimized)

$\displaystyle f(x,y)=2x^2 + 4xy$

and a constraint function

$\displaystyle g(x,y)=x^2y=9$

I make a Lagrangean:

$\displaystyle lagrange(x,h,\lambda)=2x^2+4xy-\lambda(x^2y-9)$

which obviously turns into

$\displaystyle L(x,h,\lambda)=2x^2+4xy-\lambda x^2y-9\lambda$,

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

$\displaystyle \frac{\partial L}{\partial x}=4x+4y-2 \lambda y = 0$

$\displaystyle \frac{\partial L}{\partial y}=4x - \lambda x^2 = 0$

$\displaystyle \frac{\partial L}{\partial \lambda}=-yx^2-9 = 0$

Is there anything obviously messed up so far? When I put the system of equations into CAS, it gives "false" as solution, so I kind of guess that I am messing something up.