I) Consider the equation G(x,y) =2 - 3xy + 3 - 7=0 1. Are the conditions of the implicit function theorem satisfied at the point (ro, yo) = (4, 3)? Justify your answer. 3 marks 2. Find the value of gy at (To, 30) - (4, 3). 2 marks II) Consider a profit-maximizing firm that uses a single input x at a cost of w dollars per unit to produce a single output via a production function y = f(x) = vx. p is the price of a unit of output. 1. Write the first-order condition for the profit-maximizing problem and find the optimal choice of the input, x* - I(p, w). 2 marks 2. Evaluate the sign for the expressions and spective economic meanings. aw and provide their re- 3 marks III) Consider the following three equations Ty W 0 y - 3z 1 + 2- 2zw 0 1. Determine the total differential of the system. 2 marks 2. Represent the total differential of the system in matrix form JV = Udz, where J is the Jacobian matrix , V = (dr dy dw) and U a vector. 2 marks 3. Using the Cramer's rule, find an expression for ag at (r, y, w; w) = (1, 4, 1, 1). 1 mark