Problem A $(8$ points$)$

Consider a firm selling two products $1$ and $2$ in two different markets. To be specific, the firm sells product $1$ in a perfectly competitive market, where the price $p_1$ is $$180$ per unit, and product $2$ in a monopolistic market, where the demand function is $p_2=130-4q_2,$ with $q_2$ representing the output in the monopoly market. The total cost function is given by $C(q_1,q_2)=3(2q_1+q_2{)}^2,$ where $q_1$ is output in the competitive market.

$($i$)$ Formulate the firm's profit function $$ as a function of $(q_1,q_2).$

$($ii$)$ Determine the critical point of the function $.$

$($iii$)$ Does the function $$ achieve the highest profit at the critical point? Explain your answer.