= 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 gives
$$
\delta F=\int_{\mathbb R}[\eta Ly+yL\eta]dx=2\int_{\mathbb R}\eta Ly\,dx,\qquad\delta G=2\int_{\mathbb R}wy\eta\,dx.
$$
For $\Lambda=F/G$, the quotient rule therefore gives
$$
\delta\Lambda=\frac2G\int_{\mathbb R}\eta(Ly-\Lambda wy)dx.
$$
Vanishing for arbitrary variations requires
$$
\boxed{Ly=\lambda wy,\qquad\lambda=\Lambda[y].}
$$
Conversely 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 $y\to0$ 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.
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