 Solution

Take real admissible functions with finite $F,G$, positive $G$, and a self-adjoint domain for the Sturm-Liouville operator. Compactly supported variations $\eta$ avoid any boundary contribution at infinity. Formal self-adjointness givesFor $\Lambda =F/G$, the quotient rule therefore givesVanishing for arbitrary variations requiresConversely an eigenfunction in the self-adjoint domain makes this variation zero. Multiplying its equation by $y$ and integrating also shows directly that its stationary quotient is $\boxed{\Lambda [y_n]={\lambda }_n}$.

Thus the stationary values are the admissible eigenvalues. Under the usual positive-weight, semibounded hypotheses the lowest is the global minimum, while higher eigenfunctions generally give saddle points. The assumption $y\to 0$ alone is not a complete domain specification for arbitrary $\rho ,\sigma$: the required integrals and boundary form must exist. The rapidly decaying oscillator trials in part (b) meet these conditions.