Answer
$C^{\prime}(x) =4,\quad R^{\prime}(x)=8-0.002x \quad P^{\prime}(x)=4-0.002x$
The profit peaks at production level of x= 2000.
Work Step by Step
Profit function: $P(x)=R(x)-C(x).$
Marginal cost function: $C^{\prime}(x)$
Marginal revenue and profit functions: $R^{\prime}(x)$ and $P^{\prime}(x)$
What it means when the marginal profit is zero is described on p. 802:
Solving $P^{\prime}(x)=0$ for x gives the exact value of x, if such a value exists, for which the profit peaks (neither increases nor decreases) with respect to production level.
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Marginal cost function: $C^{\prime}(x) =4$
Marginal revenue function:
$R^{\prime}(x)=8-0.001(2x)=8-0.002x$
Marginal profit function$: $
$P^{\prime}(x)=8-0.002x-4=4-0.002x$
$P^{\prime}(x)=0$
$4-0.002x=0$
$4=0.002x\qquad\div 0.002$
$x=\displaystyle \frac{4}{0.002}=2000$
The profit peaks at production level of x= 2000.
