17. Suppose you want to write a program that will collect information on a customer's tastes and customize web content accordingly. By monitoring online shopping habits, you are able to collect a set of pairwise preferences on a set X of products. If a, y EX are two different products, we say that x y if the customer prefers y over a. (In order to satisfy the reflexive property, we stipulate that a 3 a for all a E X.) Suppose you know the following things about your customer. Customer prefers: Over: lettuce broccoli cabbage broccoli cabbage tomatoes carrots cabbage carrots lettuce asparagus lettuce mushrooms tomatoes corn tomatoes corn carrots eggplant eggplant carrots asparagus onions mushrooms 8 onions corn In order for (X,<) to be a poset, we must also assume that the customer's preferences are transitive. (a) Draw the Hasse diagram for (X,3). (b) What is/are the customer's favorite vegetable(s)? (I.e., what are the maximal element(s)?) What is/are the least favorite? (c) Use topological sorting to rank order these vegetables according to the customer's preferences. Is the ranking unique?

Section 3.2: HG #6) verity that the function Satisfiees the three hypotheses of kolle's theorem on the given interval. Then find all numbers c that satisfy the conclusion of ROlle's Theorem. fax)=x3-2x²-4x +2, [-2,2] to

Linear Algebra Question: Suppose x1, x2, x3 are 3-tuples in R^3 satisfying the following: a. x1 is orthogonal x2. b. x2 is orthogonal to x3. Must it be the case that x1 is orthogonal x3? If so, prove the claim. If not, explain why, and give a counterexample.

Q4 Revenue equivalence Ill  Consider a sealed-bid auction in which the highest-bidder wins and pays the amount that they bid and the lowest bidder receives a payment equal to the difference between the first- and second-highest bids. Does this auction pay the seller the same revenue as the four standard auctions? (That is, does the revenue equivalence theorem apply here?) Why or why not? Be specific.