$(25$ points$)$ In Saint Paul, there is only one producer of $$clairs$,$ who acts as a monopoly. The inverse demand function for an $$clair is given by $P=12-\frac{Q}{2},$ where $P$ is the price per $$clair and $Q$ is the quantity demanded for $$clairs$.$ Suppose the cost function of this producer is given by $C(Q)=4+2Q$

$($a$)(5$ points$)$ Compute the marginal cost and average cost functions, as well as the marginal revenue function.

$($b$)(5$ points$)$ Compute the profit maximizing output, price and profit.

$($c$)(5$ points$)$ Compute the price elasticity at profit maximizing output.

$($d$)(5$ points$)$ Suppose now that the cost function has changed and is now given by $C(Q)=4+Q^2.$ Compute the marginal cost and average cost functions, as well as the marginal revenue function.

$($e$)(5$ points$)$ Compute the firm's profit$-$maximizing output, price, and profit based on her new cost function.