In a brief review of thermodynamics, our lecture notes read
Thermodynamic potentials are concave in their extensive variables and convex in their intensive variables.
Alright, we start with $U(S,V)$, with both its proper variables being extensive. Now all other thermodynamic potentials that we can derive by means of the Legendre transform are intensive in the new, intensive variables. By the properties of the transform, the transformed function is concave in the new variable, if it was convex in the old variable, and vice-versa.
But can I show that $S(U,V)$ is concave in its proper variables, and $U(S,V)$ is convex in its proper variables and that $S$ is maximal for the system in equilibrium with only the formulation of the 2$^{\text{nd}}$ Law of Thermodynamics
In an isolated system ($\delta Q=0$), the entropy can only increase.
or do I need an additional "though experiment" or postulate for that?