Given that the demand function of a monopolist is q=9-1/8 p and its total cost function is C(q)=1/3 q^3-4.5q^2+12q+18. find the maximum profit of the monopolist

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The price function is p = 8(9-q)

The sales function is price x quantity sold = pq = 8q(9-q) = S(q)

The profit function is given by

R = S(q) - C(q)

where C(q) is the cost function

If the cost function is given by

C(q) = 1/3q^3 - 4.5q^2 + 12q + 18

then

R(q) = 8q(9-q) - 1/3q^3 + 4.5q^2 - 12q -18

       = 72q - 8q^2 - 1/3q^3 + 4.5q^2 - 12q - 18

Gathering terms

R(q) = -1/3q^3 + (4.5 - 8)q^2 + (72 - 12)q - 18

       =...

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