A drug is injected.... When is the concentration highest?

[kimmy_koo51]

A drug is injected into the body in such a way that the concentration, C, on the blood at the time, t, hours is given by the function C(t) = 10 (e^-2t - e^-3t). At what time does the highest concentration occur between one and five hours?

C(1) = 0.855
C(5) = 0.000451

C'(t) = 10 (-2e^-2t + 3e^-3t)
C'(t) = 10 (e^-2t) (-2 + 3e^-t)

0 = e^-2t
Inadmissible

0 = -2 + 3e^-t
2/3 = e^-t
ln 2/3 = t
0.41 = t

I got stuck here because if I recall correctly during optimization problems like this you check to make sure if x (or in this case t) is in the range. The answer I got here is out of range. Does this make the maximum concentration be at 1 hour or did I do something wrong.

 

[Unco]

Your work is excellent, Kimmy (though, in future, type "e^(-2t)", etc). You have correctly found that the absolute maximum over the given domain occurs at an endpoint (t=1), as indeed the relative maximum occurs outside this interval.