= Solution
Take $f$ as the tensile <force> exerted on the band, so reversible stretching supplies work $f\,dx$. At fixed other mechanical variables,
$$
dU=T\,dS+f\,dx,\qquad dF=-S\,dT+f\,dx.
$$
Equality of mixed <derivatives> of the <Helmholtz free energy> yields the <thermoelastic Maxwell relation for a stretched band>
$$
\boxed{\left(\frac{\partial S}{\partial x}\right)_T=-\left(\frac{\partial f}{\partial T}\right)_x.}
$$
For 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.
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