A price-taker firm has a production function f (L, K) = L^(1/3) + 2K^(1/3). Let w = $1 be the unit price of labor input L, and r = $2 be the unit price of capital input K. The market price of firm's output is p = $12. (a) Find the firm's profit maximizing L, K and q. (b) Find the cost-minimizing L(q) and K(q), to produce q units, and the cost function c(q). (c) Suppose you maximize the profit function pq c(q), using the cost function you found in part (b), show that the profit maximizing quantity is the same as in what you found in part (a)