 Solution

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 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.

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.