Afternoon everyone. I am currently doing an investigative task on the surge function - however it is being modelled by C(t) = A * t^p * e^-bt. (As opposed to At * e^-bt). The surge function models the drug level (in ug/L) of a patient after t hours after consuming a drug. The same patient then continues to consume the drug every 6 hours afterwards. The minimum drug level required for the drug to be effective is 2ug/L. I want to find the relationship between a, p and b such that once the drug passes the 2ug/L threshold, it remains above it throughout the entire time.
My progress: Let D(t) be the total drug level at time t, and C(t) be the function that models the consumption of 1 drug. Given C(t) approaches 0 as t approaches infinity, we know that each subsequent drug intake will result in higher results than the last. eg D(t) at 6<t<12 equals C(t) + C(t-6). Therefore we only need to find the lowest point inbetween the first and second drug.
The maximum point of C(t)(from my math) occurs at t = p/b. Therefore if p/b is less than 6, the drug levels are always growing within the 6 hours, and therefore D(t) is always growing and therefore never drops below 2.
There are two other cases: when the min point is at 6 hours, and when the min point is a local stationary point after 6 hours. My CAS won't solve when the derivative equals 0 so it can't find a stationary point in terms of the unknowns.
Overall I am stuck, and would appreciate any suggestions. It may be true that this is unsolvable, as this is an investigative task and may not have a real solution. Thanks all