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).