When I try to find extrema of entropy for a pdf p(x), I use a lagrange mulitplier to impose the normalizaiton constraint, grind through the steps, and I get that p(x) is constant. This makes sense, since a flat distribution maximizes the entropy. But why does this method not also give me the function that minimizes the entropy? I assume this would be a delta function, yes? So why does that answer not fall out? I can only guess that the delta function is a sort of "boundary" of the variation space. When dealing with functions, the derivative might not be zero at an extrema that lies on a boundary... Is something analogous to that the reason the variational method I applied doesn't work to minimize entropy?