I have posted this problem in the applied math section. Curious to hear what people here would say.

I have a (maybe trivial?) question on constrained optimization. Assume that I have the following maximization problem:

$\displaystyle
\max_{x,y} -2x^{2}+3xy-3y^{2} -10x-10y,
$
subject to
$\displaystyle
-2x^{2}+3xy-10x \geq 0.
$

I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier $\displaystyle \lambda$.
$\displaystyle
-4x + 3y - 10 + \lambda (-4x + 3y -10) = 0,
$
$\displaystyle
3x -6y - 10 + \lambda (3x) = 0.
$

By the Kuhn Tucker conditions, we know that if $\displaystyle \lambda > 0$, then the constraint is binding. However, from the first order conditions, we can see that the multiplier $\displaystyle \lambda$ is negative! How can this be? Am I doing things wrong here?

I know that this could mean that the constraint is always non-binding. But what if we changed the constraint to
$\displaystyle
-2x^{2}+3xy-10x \geq 100.
$

This clearly has an effect on the problem. So does this mean that the regularity condition (constraint qualification) is violated?

Please help!
Thank you