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;

$\max _{q}q\cdot p(q)-c(q)$
$\max _{p}p\cdot q(p)-c(q(p))$