4. (25 PTs) Consider a consumer whose preference is given by the utility function u(x1x2) = x1 + x2. The consumer has an income of w and faces prices p and 1 for goods 1 and 2 respectively. Assume p at 1. Goods cannot be consumed in negative amounts and the consumer need not spend all his income. (i) (3 PTS) Write down the optimisation problem of the consumer along with any non-negativity restrictions. (ii) (4 PTS) Prove whether the two conditions for Kuhn-Tucker's theorem satised. (iii) (9 PTS) Write down the Lagrange function and nd the critical points of this optimisation problem. (iv) (4 PTS) Is the critical point you found a global maximum? Prove or disprove. (v) (5 PTS) Write down the demand function for good 1 in terms of p and draw it in a graph. Is this function differentiable? Please answer the following questions, thank you