Question

If \(p\left( x \right)\) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is

\(A\left( x \right) = \frac{{p\left( x \right)}}{x}\)

(a) Find \(A'\left( x \right)\). Why does the company want to hire more workers if \(A'\left( x \right) > 0\)?

(b) Show that \(A'\left( x \right) > 0\) if \(p'\left( x \right)\) is greater than the average productivity.

Short answer

(a) The average productivity of the workforce increases as its size increases.

(b) It is proved that \(A'\left( x \right) > 0\) if \(p'\left( x \right)\) is greater than the average productivity.

Step-by-step solution

Average rate of change

The average rate of changeof \(y\) in relation to \(x\) over the interval \(\left( {{x_1},{x_2}} \right)\) is the difference quotientas shown below:

\(\frac{{\Delta y}}{{\Delta x}} = \frac{{f\left( {{x_2}} \right) - f\left( {{x_1}} \right)}}{{{x_2} - {x_1}}}\)

And which could be regarded as the slope of the secant line PQ.

Determine \(A'\left( x \right)\) and explain why the company want to hire more workers

The average productivity of the workforce is \(A\left( x \right) = \frac{{p\left( x \right)}}{x}\).

Use quotient rule to evaluate \(A'\left( x \right)\) as shown below:

\(\begin{aligned}A'\left( x \right) &= \frac{d}{{dx}}\left( {\frac{{p\left( x \right)}}{x}} \right)\\ &= \frac{{xp'\left( x \right) - p\left( x \right) \cdot 1}}{{{x^2}}}\\ &= \frac{{xp'\left( x \right) - p\left( x \right)}}{{{x^2}}}\end{aligned}\)

When \(A'\left( x \right) > 0\) then \(A\left( x \right)\) is increasing. In other words, the average productivity of the workforce increases as its size increases.

 Show that \(A'\left( x \right) > 0\) if \(p'\left( x \right)\) is greater than the average productivity

When \(p'\left( x \right)\) is greater than the average productivity, then;

\(\begin{aligned}p'\left( x \right) > A\left( x \right)\\p'\left( x \right) > \frac{{p\left( x \right)}}{x}\\xp'\left( x \right) > p\left( x \right)\\xp'\left( x \right) - p\left( x \right) > 0\\\frac{{xp'\left( x \right) - p\left( x \right)}}{{{x^2}}} > 0\\A'\left( x \right) > 0\end{aligned}\)

Thus, it is proved that \(A'\left( x \right) > 0\) if \(p'\left( x \right)\) is greater than the average productivity.