A competitive firm has an average variable cost function given by AVC $[$y$]$

$=10+4y.$

Moreover, when the market price is $$42,$ the firm's maximum profit is $$14.$

Given this information:

$($a$)$ Derive the firm's cost function c$($y$),$ marginal cost, MC$($y$),$ and average total

cost$,$ ATC$($y$).$ Show MC$($y$),$ AV C$($y$),$ and ATC $($y$)$ on a graph.

$($b$)$ Let the market price be $$82.$ How much will the firm produce? Determine the

associated profit and producer surplus. Show your results in the graph you

constructed for part $($a$).$