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 celebrated relation between entropic small changes and mean energy ones, [Formula: see text], does not seems 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.