Please provide an explanation for the question, thanks in advance.

[f Carla maximizes her utility function U(x1, x2) = 2.1:"; + 411an subject to 13111 -|- pzxz = m, where :1 is cheese and 12 is bread. a) Obtain the Engel curve of good x1. Is the slope of the Engel curve positive? And what is the type of good Kl? b) Given that p1 is LEIO, p2 is LE6 and Carla's income is LEIZUO, if the producer of bread decided to raise the price of his product to LE8 to increase his revenue. Is his expectation rational? Show your answer mathematically and interpret it. c) What is the consumer surplus change after the increase of the price of bread to LE8? (Hint: it is better to use a simple integral we saw in class, since using the area ofa triangle will give you an approximate answer.) d) Find another utility function representing the same preference and prove it mathematically. e) Suppose that cheese prices increase by 10%, and the total quantity of groceries in the market increases by 8%. What is the elasticity of supply of groceries? Interpret your answer in the context of a 1% change in groceries prices