A Los Angeles firm uses a single input to produce a recreational commodity according to a production function f (x) = 4√x, where x is the number of units of input. The commodity sells for $100 per unit. The input costs $50 per unit.

a. Write down a function that states the firm’s profit as a  function of the amount of input.

b. What  is the profit-maximizing amount of input? 16. of output? 16. How much profits does it make when it maximizes profits? 

c. Suppose  that the firm is taxed $20 per unit of its output and the price of its input is  subsidized by $10. What is its new input level? 

d. Suppose  that instead of these taxes and subsidies, the firm is taxed at 50% of its  profits. Write down its after-tax profits as a function of the amount of input. What is the profit-maximizing amount of output? How much profit does it make after taxes?