  Davidson potentials of the form $\beta^2 +\beta_0^4/\beta^2$, when used in
the E(5) framework, bridge the U(5) and O(6) symmetries, while they bridge the
U(5) and SU(3) symmetries when used in the X(5) framework. Using a variational
procedure, we determine for each value of angular momentum $L$ thevalue of
$\beta_0$ at which the rate of change of various physical quantities (energy
ratios, intraband B(E2) ratios, quadrupole moment ratios) has a maximum, the
collection of the values of the physical quantity formed in this way being a
candidate for describing its behavior at the relevant critical point. Energy
ratios lead to the E(5) and X(5) results (whice correspond to an infinite well
potential in $\beta$), while intraband B(E2) ratios and quadrupole moments lead
to the E(5)-$\beta^4$ and X(5)-$\beta^4$ results, which correspond to the use
of a $\beta^4$ potential in the relevant framework. A new derivation of the
Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product.
