3. [9 Marks] Consider a firm with production function f( 21, z2) - 23 2 . The prices of inputs z1 and z2 are denoted as w1 > 0 and w2 > 0, and the output price is denoted as p > 0. (a) (2 marks) Write down the Lagrange function for the profit maximisation problem of this firm and derive the first order conditions for an interior solution. (b) (2 marks) Under what circumstances can there exists an optimal production plan with q* = 0 and z1 = 25 - 0? Verify if these circumstances are met. (c) (2 marks) Under what circumstances can there exists an optimal production plan with q* > 0 and z2 - 0 and/or an optimal production plan with q* > 0 and z* = 0? Verify if these circumstances are met. (d) (3 marks) Derive the firm's supply function, q(w1, w2, p), and the firm's demand functions for factors of production, z1 (w1, w2, p) and z2(w1, w2, p)