= Solution
Take nonzero real trial functions $y$ with $y,y'$ <square-integrable> and sufficient decay for <integration by parts>. For a compactly supported <variation> $y\mapsto y+\varepsilon\eta$, the <first variations> are
$$
\delta F=2\int_{\mathbb R}(y'\eta'-V_0e^{-x^2}y\eta)\,dx=2\int_{\mathbb R}(-y''-V_0e^{-x^2}y)\eta\,dx,\qquad\delta G=2\int_{\mathbb R}y\eta\,dx.
$$
The <quotient rule> for the <functional> then yields
$$
\delta\Lambda=\frac2{G[y]}\int_{\mathbb R}\left[-y''-V_0e^{-x^2}y-\Lambda[y]y\right]\eta\,dx.
$$
By the <fundamental lemma of the calculus of variations>, stationarity for all such $\eta$ is equivalent to
$$
-y''-V_0e^{-x^2}y=\Lambda[y]y.
$$
Conversely, multiplying a decaying <eigenfunction> equation by $y$ and using <integration by parts> gives $F[y]=EG[y]$, so its <Rayleigh quotient> equals $E$ and its <first variation> vanishes. Thus the stationary values of the <functional> are the energy <eigenvalues>. The word "extrema" in the question includes stationary points here: higher <eigenfunctions> can be saddle points of the unrestricted <Rayleigh quotient>.
The <Rayleigh-Ritz variational principle> gives
$$
\boxed{E_0=\inf_{y\ne0}\Lambda[y],\qquad E_0\leq\Lambda[y_T].}
$$
Choose an admissible trial family, calculate its <Rayleigh quotient>, and minimize over its parameters to obtain the best upper bound within that family. Equality holds for the exact <ground state>; a restricted family generally gives only an upper bound.
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