Differentiation
Profit for a monopolist`s product, the cost function is
c=0.004q^3(to the power 3) + 20q + 5000
and the demand function is
p=450 - 4q
Find the profit-maximizing output
profit is given by pq-c=(450 - 4q)q-0.004q^3 - 20q - 5000 = 0.004q^3-4q^2+430q-5000
Differentiate with respect to q and get
0.012q^2-8q+430.
Now set this equal to 0 and apply the quadratic formula,
q=(8-sqrt(8²-4*0.012*430))/(2*0.012) or q=(8+sqrt(8²-4*0.012*430))/(2*0.012).
The second isnt possible as there p<0, so your answer is q=(8-sqrt(8²-4*0.012*430))/(2*0.012)