I'm going through Callen's Thermodynamics book. After demonstrating from a physical point of view the maximum entropy implies minimum energy, the inverse argument is left as an exercise:
Show that if the entropy were not maximum at constant energy then the energy could not be minimum at constant entropy.
Hint: First show that the permissible increase in entropy in the system can be exploited to extract heat from a reversible heat source (initially at the same temperature) and to deposit it in a reversible work source. The reversible heat source is thereby cooled. Continue the argument.
The minimum energy principle states that for a given entropy, the system will reach equilibrium at minimum energy. Now if we look at the whole system, including the heat and work sources, transferring heat from a reversible source will neither change the energy of the system, nor the entropy (since $ds=\frac{dQ}{dT}$, and $T_1=T_2$). Storing the energy in the work will yet cause no change in the total energy, and since it's reversible neither in the entropy as well. The heat source will be cooled by $\frac{dQ}{C}$. Yet, I can't see where is it taking us.