$($a$)$ The monopolist faces a demand curve given by D$($p$)=100$ p$.$ Its cost function is c$($y$)=2$y$.$ What is its optimal level of output and price?

$($b$)$ A firm has a supply function given by S$($p$)=2$p$.$ Its fixed costs are $50.$ If the price changes from $10$ to $20,$ what is the change in its profits?

$($c$)$ If average variable costs exceed the market price, what level of output should the firm produce? What if there are no fixed costs?

$($d$)$ If D$($p$)=100/$p and c$($y$)=$ y$^2,$ what is the optimal level of output of the monopolist?