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

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

$\backslash$table$[[$revenue$,$$$,],[$profit$,$$$,],[\backslash$table$[[$marginal revenue$],[$marginal profit$]],$$$,$per additional copy$],[$per additional copy,,$]]$

Interpret the results.

The approximate $]$ 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 $[$