Your college newspaper, The Collegiate Investigator, sells for $90$ per copy. The cost of producing x copies of an edition is given by

C$($x$)=60+0\mathrm{.}10$x $+0\mathrm{.}001$x$2$ dollars.

$($a$)$ Calculate the marginal revenue R$\text{'}($x$)$ and profit P$\text{'}($x$)$ functions. HINT $[$See Example $2.]$

R$\text{'}($x$)=$

P$\text{'}($x$)=$

$($b$)$ Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold $500$ copies of the latest edition.

revenue $

profit $

marginal revenue $

per additional copy

marginal profit $

per additional copy

Interpret the results.

The approximate

$---$Select$---($profit or loss$)?$

from the sale of the $501$st copy is $

$.$

$($c$)$ For which value of x is the marginal profit zero?

x $=$

copies

Interpret your answer.

The graph of the profit function is a parabola with a vertex at x $=\_\_\_\_\_\_$

$,$ so the profit is at a maximum when you produce and sell $\_\_\_\_\_\_\_\_\_\_$

copies.