= Solution
\b[A gives the same optimum:] replacing $\alpha$ by $2\alpha$ merely reparameterizes the same set of positive-width <Gaussian functions>, so its best bound is unchanged.
\b[B is worse as a ground-energy estimate:] it has odd parity and a node at the origin, whereas the <ground state> of this even confining one-dimensional potential is even and nodeless. It is therefore <orthogonal> to the exact <ground state> and belongs to the odd sector, whose lowest <energy> is already above the ground <energy>; the extra factor $x$ cannot be removed by varying its width. A separate numerical minimization gives a best value about $1.92372$ for B, well above the original <Gaussian function> bound.
\b[C can improve the bound:] setting $\beta=0$ recovers every original <Gaussian function>, while nonzero $\beta$ adds shape freedom. Minimizing over this larger family can never give a worse optimum, and the additional freedom does lower the quartic-oscillator estimate. Inclusion of a family alone proves “no worse”; improvement is not guaranteed for every conceivable potential, for example one already solved exactly by a <Gaussian function>.
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