Analyzing Monthly Sales

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.

1. When are the monthly sales increasing? When are they decreasing?
2. Find the rate of change of sales.
3. When is the rate of change of sales \(R^\prime\) increasing? When is it decreasing?
4. When is the rate of change of sales a maximum?
5. Find any inflection points of \(R(t)\).
6. Interpret the result found in (e).

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.