  We use density matrix renormalization group to study the first-order quantum
phase transition induced by a magnetic field $h$ in a frustrated ferrimagnetic
chain. The magnetization ($m$) curve as a function of $h$ presents a
macroscopic jump and the energy curve as a function of $m$ has two global
minima. We characterize the two competing phases and study the phase-separated
states in the coexistence region. Also, we observe that the transition is
accompanied by an increase in the number of itinerant singlet pairs between
sites in the unit cells of the chain. Finally, we identify the critical point
at the end of the first-order transition line and a crossover line.
