A monopolist sells to a market in which the fraction of buyer types whose valuation is v or less is given by F(v). The lowest valuation customer in the market is a customer with v = k, so we have F(k) = 0. The monopolist has constant marginal and average cost of production, denoted by c. Assume that k is greater than or equal to c.
Suppose that it is optimal for the monopolist to serve the entire market (i.e. the monopoly price equals k). Derive a necessary first order condition for profit maximization.