A tranquilizer is injected into a muscle from which it enters the bloodstream.

The concentration \(C\) in \({\text{mg}}{{\text{l}}^{ - 1}}\), of tranquilizer in the bloodstream can be modelled by the function \(C(t) = \frac{{2t}}{{3 + {t^2}}},{\text{ }}t \ge 0\) where \(t\) is the number of minutes after the injection.

Find the maximum concentration of tranquilizer in the bloodstream.

use of the quotient rule or the product rule     M1

\(C'(t) = \frac{{(3 + {t^2}) \times 2 - 2t \times 2t}}{{{{\left( {3 + {t^2}} \right)}^2}}}\;\;\;\left( { = \frac{{6 - 2{t^2}}}{{{{\left( {3 + {t^2}} \right)}^2}}}} \right)\;\;\;{\text{or}}\;\;\;\frac{2}{{3 + {t^2}}} - \frac{{4{t^2}}}{{{{\left( {3 + {t^2}} \right)}^2}}}\)     A1A1

Note:     Award A1 for a correct numerator and A1 for a correct denominator in the quotient rule, and A1 for each correct term in the product rule.

attempting to solve \(C'(t) = 0\;\;\;{\text{for }}t\)     (M1)

\(t =  \pm \sqrt 3 \;\;\;{\text{(minutes)}}\)     A1

\(C\left( {\sqrt 3 } \right) = \frac{{\sqrt 3 }}{3}\;\;\;\left( {{\text{mg}}{{\text{l}}^{ - 1}}} \right)\;\;\;{\text{or equivalent.}}\)     A1

[6 marks]

This question was generally well done. A significant number of candidates did not calculate the maximum value of \(C\).