Final answer:
The maximum change in population over 5 time periods is 75 units from the original population of 75. Thus, the population N(5) must be between 75 and 150, inclusive.
Step-by-step explanation:
The question involves differential calculus, specifically dealing with the rate of change of a population over time. We're given the initial population size, N(0) = 75, and a bound on the absolute rate of change, |dN/dt| ≤ 15, over a certain time interval, [0, 5]. This information implies a linear growth model where the population size cannot change at a rate faster than 15 units per time period.
To estimate the population at time t = 5, we could use the fact that the rate of change does not exceed 15. Since the population can change at most by 15 units each time period, after 5 time periods, the population size can increase or decrease by at most 15 × 5 = 75. Therefore, the population at time 5, denoted by N(5), must be within 75 units of the original 75. Hence, it would be between 0 and 150. Considering only positive populations, N(5) can be anywhere from 75 to 150, inclusive.