$2.$ A firm has a cost function that satisfies the following: $($i$)$ PS$($y$)\backslash$Pi $($y$)=2$ for all y$,($ii$)$ SMC$(0)=$AVC$(0)=1$ and $($iii$)$

$$y

$$AVC$($y$)$

$$

$=$

$2$

$1$

$$

$(20$pts$)$ a$)$ Derive the firm's cost function C$($y$),$ marginal cost function SMC$($y$),$ average cost SAC$($y$),$ and average variable cost function AVC$($y$).$ Show AVC,SMC and SAC on a graph. b$)$ Suppose the market price is $11.$ How many units would the firm produce? Determine the associated profit and producer surplus for the firm.