A physical point of view on the arithmetic and geometric mean inequality

We propose a simple experiment designed to justify the arithmetic and geometric mean inequality by means of the laws of thermodynamics. The experiment consists in measuring the entropy variation Delta S in the thermodynamic irreversible process of cooling a metal in water. By considering the metal and water as a single isolated system, the arithmetic and geometric theorem is seen to hold by noticing that Delta S is positive for this irreversible transformation. These interdisciplinary activities may be used to reinforce basic concepts in thermodynamics in high school or first-year college students.