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:

subject to

I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier .

By the Kuhn Tucker conditions, we know that if , then the constraint is binding. However, from the first order conditions, we can see that the multiplier 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

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

Please help!

Thank you