= Solution
The radial <Rayleigh quotient> is
$$
k_{\rm trial}^2=\frac{\int_0^ar^2(\psi'_{\rm trial})^2dr}{\int_0^ar^2\psi_{\rm trial}^2dr}.
$$
For the given trial profile, the numerator is $4a^{-4}\int_0^ar^4dr=4a/5$. The denominator is
$$
\int_0^ar^2(1-r^2/a^2)^2dr=a^3\left(\frac13-\frac25+\frac17\right)=\frac{8a^3}{105}.
$$
Therefore
$$
\boxed{k_0^2\leq k_{\rm trial}^2=\frac{21}{2a^2}=\frac{10.5}{a^2}.}
$$
The profile is regular at the origin and vanishes at the wall, so it belongs to the admissible trial space. This is the variational estimate for the <radial Dirichlet modes of a spherical cavity>.
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