Advanced Microeconomics/Monopoly Pricing

A monopolist produces a good with the demand function $$q(p)$$ which (by assumption) has an inverse $$ p(q) $$ and costs given by $$c(q)$$. Since it supplies the entire market, the monopolist simultaneously chooses output and price. Thus, the profit maximization problem may be written two ways; $$ \max_{q} q\cdot p(q) - c(q) $$ $$ \max_{p} p\cdot q(p) - c(q(p))$$