A young boy, who is a budding entrepreneur, starts his career by selling newspapers. he wants to know how many newspapers to order from the wholesaler every day to maximize his expected net profit. He buys newspapers for $1 each and sells them at $1.50 each, making a cool profit of $0.50 per newspaper sold. For every unsold newspaper he loses $1. If the demand exceeds his stock newspapers, then he incurs a penalty cost of $0.75 for each extra newspaper because of customer dissatisfaction. The demand for newspapers is uncertain and follows the distribution given below: (Demand (x), Probability (f(x)): (1, 0.01), (2, 0.04), (3, 0.05), (4, 0.10), (5, 0.15), (6, 0.20), (7, 0.20), (8, 0.10), (9, 0.10), (10, 0.05).

a.) Find the expected demand.

b.) Let n denote the number of newspapers that the boy orders. Write the net profit as a function of n and random demand X. (Hint: the profit function has different formulas for X less than or equal to n and X>n)

c.) Find the value of n that maximizes the expected net profit. you may limit your search for optimum n to n= 5, 6, and 7.