An experimental garden plot contains N annual plants, each of which produces S seeds that are dropped within the same plot. A botanical model measures the number of offspring plants A(N) that survive until the next year by the function

where c and p are positive constants.

a. For what value of N is A(N) maximized? What is the maximum value? Express your answer in terms of S, c, and p.

b. For what value of N is the offspring survival rate A'(N) minimized?

c. The function

is called the net reproductive rate. It measures the number of surviving offspring per plant. Find F'(N), and use it to show that the greater the number of plants, the lower the number of surviving offspring per plant. This is called the principle of density-dependent mortality.

A(N) NS 1 + (CN)P