The function S(t)=t^2e−0.1t gives the level of medication (in μg) in a patient's bloodstream t minutes after a dose is given. CONTINUED IN DESCRIPTION

Question

Use the first and second derivative tests to to determine the local extrema and inflection points for this function...including where the function is increasing, decreasing, concave up, and concave down. Use a limit to determine what happens to the drug level long term.

Answer 1

Given S(t) = t²e - 0.1t (Note this is based on the question you wrote down, the function may be incorrect based on your notation).

The first and second derivatives are S'(t) = 2et - 0.1 and S"(t) = 2e, respectively.

To find critical points set S'(t) = 0, this leads to the equation: 0 = 2et - 0.1, solving for t yields t = 0.1/(2e). Since the second derivative is constant at a value of 2e (which is greater than 0), the critical value is a minimum.

There are no inflection points in the function given and the concavity is always up, which makes absolutely no sense since this problem refers to the amount of medication in the bloodstream at any time t after medication is administered. This medicine is showing a negative concentration in blood and then it is increasing to infinity as time goes to infinity. The function is non-sensical since most medication is metabolized and slowly disappears from your bloodstream.