Question

The quantity of a drug in the bloodsteam t hours after a tablet is swallowed is given, in mg, by q(t)=40((e^-t)-(e^-2t))
What is the maximum amount of the drug in the bloodstream at one time?

Answer 1

the maximum is at
\[q \prime (t)=0 \implies 40(-e ^{-t}+2e ^{-2t})=0 \implies 2e ^{-2t}=e^-t\]
take ln for both sides, you get
\[\ln(2e ^{-2t})=\ln(e ^{-t}) \implies \ln2-2t=-t \implies t=\ln2\]
t=ln2 is the only critical point..substitute in the original function with t=ln2, you get:
q(ln2)=40(1/2 - 1/4)=10 mg