One very important question facing hospital is: How big must a hospital be (in terms of patient-days of care) to minimize the cost per patient-day? According to one well-known study, the total cost (in dollars) of operating a hospital (of a particular type) can be approximated by: C = 4,700,000 + 0.00013X2, where X is the number of patient-days. 

a) Derive a formula for the relationship between cost per patient-day and the number of patient-days. 

b) On the basis of the results of this study, how big must a hospital be (in terms of patient-days) to minimize the cost per patient-day? 

c) Show that your result minimizes, rather than maximizes, the cost per patient-day.