The population of fish in a certain lake at time t months is given by the function:

P(t) = 20,000/(1+24e^(-t/4)) , where T is greater than or equal to 0. There is an upper limit on the fish population due to the oxygen supply, available food, etc.

A. What is the initial population of fish?

B. When will there be 15,000 fish?

C. What is the maximum number of fish possible in the lake?

A and B i know how to do, but C is a little confusing. To find the maximum, i took the derivative of P(t), which after simplifying, is 120,000 e^(-t/4) / (1+24e^(-t/4))^2. i tried to find critical numbers by setting the numerator and denominator equal to 0, but on the top i end up with e^(-t/4) = 0 and exponential functions never equal 0 so that's no solution. and on the bottom i get e^(-t/4) = -1/24 and since exponential functions are never negative, that one has no solution either. i saw the answer that said take the limit as t approaches infinite and you get 20,000 as the maximum. but how come i couldn't use the derivative to find the maximum in this problem?