Answer
(a) $A'(x) = \frac{xp'(x)-p(x)}{x^2}$
If $A'(x) \gt 0$, then the average productivity of the workforce would increase if more workers were added. Therefore, the company would want to hire more workers in order to improve the average productivity of the workforce.
(b) If $p'(x)$ is greater than the average productivity, then $A'(x) \gt 0$
Work Step by Step
(a) $A(x) = \frac{p(x)}{x}$
$A'(x) = \frac{xp'(x)-p(x)}{x^2}$
If $A'(x) \gt 0$, then the average productivity of the workforce would increase if more workers were added. Therefore, the company would want to hire more workers in order to improve the average productivity of the workforce.
(b) Suppose $p'(x)$ is greater than the average productivity.
Then:
$p'(x) \gt \frac{p(x)}{x}$
$xp'(x) \gt p(x)$
$xp'(x) - p(x) \gt 0$
$\frac{xp'(x) - p(x)}{x^2} \gt 0$
$A'(x) \gt 0$
