A certain company sells two products, one basic version and one premium version, which compete for market share. The company has found that if they charge $p_1$ dollars per unit for the basic version and $p_2$ dollars per unit for the premium version, their sales will be $s_1=400-7p_1+5p_2$ for the basic version and $s_2=1000-6p_2+3p_1$ for the premium version.

$($a$)$ Write a formula for the company's total sales across both products. Show that the partial derivative of the company's total sales with respect to the price of either product is negative $($that is$,$ the company can never increase their overall sales by charging more for one product while leaving the other product at the same price.$)$

$($b$)$ Write a formula for the company's total revenues $R(p_1,p_2)$ in terms of the prices they charge for each product.

$($c$)$ Find the values of $p_1$ and $p_2$ which yield a global maximum for the company's total revenues.

$($d$)$ Do the values you calculated in part $($c$)$ make sense? Are prices positive? Are sales positive? Does the company charge more for the premium product than for the basic product?

$($e$)$ If you have time, redo this problem but with $s_1=400-7p_1+8p_2.$ Show that this model isn't realistic $($that is$,$ show that it leads to impossible answers in part $($d$).)$

Page $2$ of $2$