  We describe in great generality features concerning constrained entropic,
functional variational problems that allow for a broad range of applications.
Our discussion encompasses not only entropies but, potentially, any functional
of the probability distribution, like Fisher-information or relative entropies,
etc. In particular, in dealing with generalized statistics in straightforward
fashion one may sometimes find that the first thermal law
$\frac{dS}{d\beta}=\beta\frac{d<U>}{d\beta}$ seems to be not respected. We show
here that, on the contrary, it is indeed obeyed by any system subject to a
Legendre extremization process, i.e., in all constrained entropic variational
problems.
