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Mathematical Method for Economics

1. The per period sales of a new product, x(t), evolves over time according to

x(t):= A / (1 + b * e - c t ) where A, b and c are positive constants.

a) By taking the limit of x(t) as t tends to infinity, show that per period sales tends to A as t increases.
b) Show that the rate of growth of sales is proportional to the difference between A and x(t).
c) Show that the maximum rate of change occurs when x(t) := A / 2
d) Find the value of t, as a function of the constants b and c, at which this maximum change occurs.