Consider a firm with a general Cobb-Douglas production function given by Y=f(K,L) = K^ a L^B where alpha (a) and beta (B) are positive constants.

(a)Suppose the firm is a perfectly competitive firm that maximizes profit.State the profit maximization problem and derive the first-order conditions.

(b)State the sufficient second-order conditions for a local maximum.Do they hold for any values of alpha and beta or only some values?What does this imply about the returns to scale of the production function?

(c)Now suppose instead that the firm has a fixed level of output that to produce and it chooses its input levels to minimize the cost of producing that output level.State the firm's choice problem in this case and derive the first-order conditions.

(d)State the sufficient second-order conditions for a local minimum.Do they hold for any values of alpha and beta or only some values?

(e)If the second order conditions for profit maximization hold, will the second order conditions for the corresponding cost-minimization problem always hold?Is the reverse true?