Question

A) C(t) = 10 (e^-0.3t - e^-0.5t), where time t is measured in hours, and concentration c(t) is in ug/ml.

b) what are the maximum and minimum concentrations of the antibiotic during the first 18 hours after injection?
c) Use the second derivative test to show that that the internal extremum at some time tv with 0 < tm < 18 is indeed a maximum.
d) Also find the time at which the inflection point occurs.

Answer 1

The question involves the use of calculus to find the maximum and minimum concentrations of an antibiotic over a time period and to determine the time at which the inflection point occurs using derivative tests.

Step-by-step explanation:

The question asks about calculating the time at which the concentration of an antibiotic, represented by a mathematical function C(t) = 10 (e^-0.3t - e^-0.5t), reaches its maximum and minimum concentrations within the first 18 hours after injection. This can be accomplished by using calculus, specifically derivative tests to find the critical points and determining whether they are maxima or minima. The second derivative test would be used to confirm the nature of these critical points, and we would need to differentiate C(t) twice to apply this test. To find the inflection point, where the concavity changes, we need to look at the second derivative and find where it changes from positive to negative or vice versa.