A firm uses two kinds of inputs, capital (K) and labor (L), to manufacture a product. The production technology is given as follows: f(K, L) = x/. One unit of capital costs 144 dollars and one unit of labor costs 16 dollars. The firm seeks to minimize the costs of manufacturing y units of the product, where y is a fixed positive number. (a) Write out formally the cost-minimization problem with which the firm is faced. (b) Use Lagrange's method to solve for the optimal input combination (K*, L*) and the associated Lagrange multiplier A\". ((2) Compute the cost function (as a function of y). The cost function is, by definition, the least possible cost of the optimal input combination that still makes it feasible for the firm to produce 3/ units of output. (d) Show that the marginal cost (the derivative of the cost function with respect to y) equals the optimal value of the Lagrange multiplier A\