Derivatives in Biology

Problem 172: Find the amount of medicine to which the body is most sensitive by finding the value of $M$ that maximizes the derivative $dR/dM$ , when

(1) $\begin{equation*} R = M^2 \left( \frac{C}{2} - \frac{M}{3}\right)\end{equation*}$

and $C$ is a constant.

Solution: We need to find the derivative with respect to $M$ twice. That is,

(2) $\begin{align*} R & = M^2 \left( \frac{C}{2} - \frac{M}{3}\right) \\ R & = \frac{CM^2}{2} - \frac{M^3}{3} \\ \frac{dR}{dM} & = \frac{2CM}{2} - \frac{3M^2}{3} \\ \frac{d^2R}{dM^2} & = C - 2M \\ 0 & = C - 2M \quad \Rightarrow \quad M = C/2. \end{align*}$

That is, by the Second Derivative Test, $M = C/2$ gives the maximum amount of medicine.

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