Question $11$

$(10$ points$)$

The demand function for a company is given by $q=8200e^{-0\mathrm{.}05p},$ where p is the selling price per item in dollars, and $q$ is the quantity of items demanded.

a$.$ Find the Revenue Function $R(p).$

b$.$ Find $R^{\text{'}}(p)$

c$.$ What price should the company charge per item to maximize revenue? Show all steps. A Sign Chart is required with work shown for test points, for full credit.

Please show work step by step