Unit monthly sales $R$ of a new product over a period of time are expected to follow the logistic function $$R=R(t) =\dfrac{20{,}000}{1+50e^{-t}}-\dfrac{20{,}000}{51}\qquad t\geq 0$$

where $t$ is measured in months.

Solution (a) We find $R^\prime (t)$ and use the Increasing/Decreasing Function Test. $$R^\prime (t) =\dfrac{d}{\textit{dt}}\left( \dfrac{20{,}000}{1+50e^{-t}}-\dfrac{20{,}000}{51}\right) =20{,}000\cdot \left[ \dfrac{50e^{-t}}{(1+50e^{-t}) ^{2}}\right] =\dfrac{1{,}000{,}000e^{-t}}{(1+50e^{-t}) ^{2}}$$

Since $e^{-t}>0$ for all $t\geq 0$, then $R^\prime (t) >0$ for $t\geq 0.$ The sales function $R$ is an increasing function. So, monthly sales are always increasing.

(b) The rate of change of sales is given by the derivative $R^\prime (t) =\dfrac{1{,}000{,}000e^{-t}}{(1+50e^{-t})^{2}},$ $t\geq 0.$

(c) Using the Increasing/Decreasing Function Test with $R^\prime$, the rate of change of sales $R^\prime$ is increasing when its derivative $R^{\prime \prime} (t) >0;$ $R^\prime (t)$ is decreasing when $R^{\prime \prime} (t) <0$. $$\begin{eqnarray*} R^{\prime \prime} (t) &=&\dfrac{d}{\textit{dt}}R^\prime (t) =1{,}000{,}000\left[ \dfrac{-e^{-t}(1+50e^{-t})^{2}+100e^{-2t}( 1+50e^{-t}) }{(1+50e^{-t})^{4}}\right] \\[12pt]&=&1{,}000{,}000e^{-t}\left[ \dfrac{-1-50e^{-t}+100e^{-t}}{(1+50e^{-t})^{3}}\right] =\dfrac{1{,}000{,}000e^{-t}}{(1+50e^{-t}) ^{3}}(50e^{-t}-1) \end{eqnarray*}$$

Since $e^{-t}>0$ for all $t$, the sign of $R^{\prime \prime}$ depends on the sign of $50e^{-t}-1$. $$\begin{eqnarray*} \begin{array}{rl@{\qquad}rl} 50e^{-t}-1 &> 0 & 50e^{-t}-1 &<0\\[4pt] 50e^{-t} &> 1 & 50e^{-t} &<1 \\[4pt] 50 &> e^{t} & 50 &<e^{t} \\[4pt] t &< \ln 50 & t &>\ln 50 \end{array} \end{eqnarray*}$$

Since $R^{\prime \prime} (t) >0$ for $t<\ln 50 \approx 3.9$ and $R^{\prime \prime} (t) <0$ for $t>\ln 50 \approx 3.9$, the rate of change of sales is increasing for the first $3.9$ months and is decreasing from $3.9$ months on.

(d) The critical number of $R^\prime$ is $\ln 50 \approx 3.9$. Using the First Derivative Test, the rate of change of sales is a maximum about $3.9$ months after the product is introduced.

(e) Since $R^{\prime \prime} (t) >0$ for $t<\ln 50$ and $R^{\prime \prime} (t) <0$ for $t>\ln 50$, the point $(\ln 50,9608)$ is the inflection point of $R$.

(f) The sales function $R$ is an increasing function, but at the inflection point $(\ln 50,9608)$ the rate of change in sales begins to decrease.