A firm operates in a perfectly competitive market whose total cost varies as TC = X3 - 3X2 - 10X + 2, and the price of the product they manufacture is given by P = 130 - 2X, where X is number of units of output.
a. What optimum number of units should be produced in order to maximize profits?
b. What is the profit if the optimum number of units is produced?
c. What is the number of units that should be produced and sold each month in order to maximize profits?
d. Show that your answer to Part (a) maximizes profit.