Multivariable Optimization: With Hessian

A monopolist sells two products x and y for which the demand functions are:

$x=50-0\mathrm{.}5P_x$

$Y=76-P_y$

And the combined cost function is

$C=3x^2+2xy+2y^2+55$

Find $($a$)$ the profit maximizing level of output $(x,y)$ for each product through optimization of the profit function,

$($b$)$ The profit maximizing price $(P_x,P_y)$ for each product,

$($c$)$ Find the maximum profit $()$ value for the monopolist, and

$($d$)$ Test the second order condition with Hessian.