  We calculate the spectrum of low-lying collective excitations in a mesoscopic
cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas as a
function of the boson-fermion repulsions. The cloud is under isotropic harmonic
confinement and its dynamics is treated in the collisional regime by using the
equations of generalized hydrodynamics with inclusion of surface effects. For
large numbers of bosons we find that, as the cloud moves towards spatial
separation (demixing) with increasing boson-fermion coupling, the frequencies
of a set of collective modes show a softening followed by a sharp upturn. This
behavior permits a clear identification of the quantum phase transition. We
propose a physical interpretation for the dynamical transition point in a
confined mixture, leading to a simple analytical expression for its location.
