The demand function of the monopolist is `q = 9 - 8*p` and the total cost function is `c(q) = q^3/3 - 4.5*q^2 + 12*q + 18` .
`q = 9 - 8*p`
=> `8p = 9 - q`
=> `p = (9 - q)/8`
The revenue earned by the firm when q products are sold is `r(q) = p*q = (q*(9 - q))/8` . The marginal revenue is `r'(q) = -(2*q-9)/8`
The total cost of the firm is `c(q) = q^3/3 - 4.5*q^2 + 12*q + 18` . The marginal cost is `c'(q) = q^2-9*q+12`
Profit is maximized when marginal cost is equal to marginal revenue; when this is the case `-(2*q-9)/8 = q^2-9*q+12` . Solving the equation gives `q = 3/2` and `q = 29/4` .
When `q = 3/2` , the profit earned is 26.15625 and when `q = 29/4` the profit earned is negative
The maximum profit earned by the firm is 26.15625