This problem appears in our calculus book. It asks to compute the maximum concentration of drug in the blood and the time when the maximum occurs as well as to find the number of inflection points and then it asks for the time at which the inflection points occur and finally the "significance" of the inflection points. The model given is $$C(t)=0.5t^2e^{-0.6t}$$ where $C(t)$ is the concentration. It was very easy to find that the maximum concentration of $\frac{50}{9}e^{-2}$ occurs at $t=10/3$ and that there are two inflection points, occurring at $t=\frac{10\pm 5\sqrt{2}}{3}$. But when the author asks for the "significance" of the inflection points I'm not sure what he means. An inflection point is a zero of the second derivative occurs. It is where the graph is "flat" (i.e. the curvature is zero) but in this case I believe the author wants the "biological" significance, and there I am stuck. Is it when the drug is being absorbed and excreted? Perhaps there is something about drug metabolism that I need to know and do not? Thank you in advance for any help!
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1why the down vote without leaving any feedback? – rog Mar 27 '17 at 09:39
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At the inflection points, the rate of variation of $C(t)$ is maximal, either positively or negatively. – Did Mar 27 '17 at 09:48
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Yes, but that isn't the biological significance. I think the author wants the inflection points to be interpreted in terms of what the drug is doing in the body, no? – rog Mar 27 '17 at 09:52
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What I mentioned does have a biological meaning, namely, that the concentration changes the fastest at these times. – Did Mar 27 '17 at 09:56
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You don't think that the question is really asking why the concentration is changing the fastest? As in what is actually going on with the drug in the system that results in this behavior of the graph? – rog Mar 27 '17 at 10:06
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What I think is that the question is asking for the meaning of the inflection points and that my previous comments answer this. – Did Mar 27 '17 at 10:09
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When "the rate of variation of $C(t)$ is maximal" you would expect maximum effect of the drug at that time $t$. For example: one has headache and takes the pain killer drug, at $t=23$ min, he will have maximum effect of the drug.
A better example: an antibiotic for infection. The guy should take tablets regularly every 8 hours and the maximum effect of the drug is at $t=4$ hours. In real research, these times should be computed in trials to be able to say how often a tablet should be taken (dosage and many other things of the tablet are important as well).
Thanks Did for answer.
Xander Henderson
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