# Given that the demand function of a monopolist is `q=9- 8p` and the total cost function is `c(q)=q^3/3-4.5q^2+12q+18` , find the maximum profit of the monopolist.

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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

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