Solve the following problem:
A hot metal of mass m, specific heat cp, and temperature T, is immersed in a cooler liquid of mass m specific heat c'p and temperature T'i The entire system is thermally insulated. If the final temperature of the metal is Tf and that of the liquid is Tj, then show by the method of Lagrangian multipliers that the condition for the entropy change of the universe to be a maximum is that Tf = Tj.