I have attached 2 screenshots, these are related with each other. One of them is giving info, another one contains questions. Could you please explain a and b with clear step by step solution? Please, do not write only answer.

Question 3. (40 marks) On constrained optimization (0,) Consider the following Optimization problem: most U = U (:13, y) subject to past: +pyy = B where pmpy and B are constants. (For your reference, the above model is very common in both Micro and Macro. This is called a standard utility maximization problem. U is the consumer's payoff or utility which is a function of the quantities of two goods :3 and y, B represents the consumers's available income (or, budget constraint) and, P95 and Py denote the prices of good a: and g, respectively.) (i) (5 marks) Write the Lagrangian and write the first order conditions (FOC) for the Lagrangian. (ii) (6 marks) Recall that Marginal rate of substitution (MRS) is defined as the ratio of marginal utilities (i.e., U. :). Show that the FOC in (i) implies that MRS should be the same as the price ratio of two goods (i.e., Px/Py). (iii) (10 marks) Now assume that U(x, y) = xy, Px = 1, Py = 4 and B = 120. Find A* along with the optimal levels of a* and y*. Calculate U*. (iv) (9 marks) Using the Bordered Hessian matrix, verify that the second-order conditions for a maximum are satisfied