Consider a Cournot duopoly where firm $1$ faces cost of the form $c_1(y_1)=y_1^2$ and firm $2$ faces

cost of the form $c_2(y_2)=2y_2^2.$ The inverse demand function is given by $P=10-2Y$ where

$Y=y_1+y_2.$

$($a$)$ State the maximization problem of firm $1$ and use the first order condition to derive firm $1\text{'}$s

reaction function.

$($b$)$ State the maximization problem of firm $2$ and use the first order condition to derive firm $2\text{'}$s

reaction function.

$($c$)$ Solve the $22$ system of reaction function to find the optimal levels of production for firm $1$

and firm $2.$

$($d$)$ Find the market quantity, the market price, and firms' profits. Show your results graphically.