Davidson potentials of the form $β^2 +β_0^4/β^2$, 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 $β_0$ 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 $β$), while intraband B(E2) ratios and quadrupole moments lead to the E(5)-$β^4$ and X(5)-$β^4$ results, which correspond to the use of a $β^4$ potential in the relevant framework. A new derivation of the Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product.