 Solution

Take $f$ as the tensile force exerted on the band, so reversible stretching supplies work $fdx$. At fixed other mechanical variables,Equality of mixed derivatives of the Helmholtz free energy yields the thermoelastic Maxwell relation for a stretched bandFor entropy-dominated rubber elasticity, increasing extension reduces configurational entropy, consistently giving a tensile force that increases with temperature at fixed extension. A force defined as the band's restoring force on its support has the opposite sign.