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