I think that if lambda is 0 then your solution happens to lie on a stationary point of the (unconstrained) objective function, and so that particular constraint has no effect. Do you agree?
Was going through a question that's already been done in the class. When I compared my results with my lecturer's I spotted I had for some critical points, whereas my lecturer only had non-zero values for it.
Is always a non-zero value? I can't be sure as I have terrible notes.