There are two kinds of Tsallis-probability distributions: heavy tail ones and compact support distributions. We show here, by appeal to functional analysis’ tools, that for lower bound Hamiltonians, the second variation’s analysis of the entropic functional guarantees that the heavy tail q -distribution constitutes a maximum of Tsallis’ entropy. On the other hand, in the compact support instance, a case by case analysis is necessary in order to tackle the issue.Instituto de Física La Plata