  We used mean-field theory based on the Bogoliubov inequality for the Gibbs
free energy to examine the magnetic properties of a mixed spin-5/2 and spin-7/2
Blume-Capel ferrimagnetic system. The thermal behaviours of the system
magnetization are classified according to the extended N\'eel nomenclature. The
system exhibits compensation phenomena where a complete cancellation of
sublattice magnetizations is observed below the critical temperature.
Temperature-dependent phase diagrams are constructed for the case of unequal
sublattice crystal field interactions. Under appropriate conditions, our
calculations reveal first-order transitions in addition to second-order ones
previously observed in Monte Carlo simulations.
