Complete analyses of optimal solutions of fed-batch fermentation are provided for recombinant product maximization for free and fixed final times and constant and variable yield coefficients when the specific product formation rate p is in the form of the Leudeking-Piret equation. In all cases, feedback forms of singular feed rates are obtained, although one or two parameters must be introduced for variable yields. The precise conditions for existence of singular feed rate are obtained and characterized in terms of the ratio of the specific growth rate of the plasmid-bearing cell (PBC) to that of the plasmid-free cell (PFC), mu(+) / mu(-), and the specific growth rate of PBC, mu(+). Thus, the inspection of the specific rates leads immediately if a singular feed rate is a part of the optimal feed rate strategy. When p is linear with respect to l and the yield coefficient is constant, the optimal singular feed rate forces the substrate concentration to start near the value at which mu(+) is maximum and moves in time in the direction of maximizing mu(+)/mu(-).