A company produces two products. The demand $(x$ and $y)$ of these products depends on

the prices $(P_x$ and $P_y.$ The demand functions are given by:

$x=-5+\frac{1}{2}{P}_x$

$y=-\frac{3}{4}+\frac{P_y}{4}$

Suppose that the company's cost function is given by:

$C(x,y)=6x-2y+7\mathrm{.}5y^2+2xy+4x^2$

$($a$)$ Find the company's revenue function $R(x,y)($Hint: see Chiang pg$.353$ for the revenue

function of a multiproduct firm$).$

$($b$)$ Hence, show that the company's profit function is given by:

$(x,y)=4x+5y-3\mathrm{.}5y^2-2x^2-2xy$

$($c$)$ Find the production levels of $x$ and $y$ at which profit is maximized.

$($d$)$ Verify that your answer for $3($c$)$ does indeed maximise profit.