Lagrange multipliers (X) measure the effect of relaxing the constraints on the optimized value of the objective function. are used in unconstrained optimization problems. are tangent to the constraint. in an optimization problem, have the effect of relaxing the constraint.

Write the answers as a fraction in reduced form.Do you know what regifting is? It is loosely defined as "giving someone a gift that was originally given to you without telling the recipient the origin of the gift." Now that you know, how often do you engage in regifting? In a survey of 5555 persons, the answers to that question were as follows. Response to the question: How often do you engage in regifting?Very often → 146 votesEvery now and then → 819 votesNot very often → 1756 votesNever → 2808 votesI'm not sure → 26 votesCurrent number of voters: 5555 If one of the persons is selected at random, what is the empirical probability of the following? (a) The person very often engages in regifting. (b) The person never engages in regifting.(c) If you get a gift and you assume that the results of the survey apply to that gift, what is the probability that you are getting a regift? Assume that the people who give regifts are the ones that did not answer "Never."

Ms. Hammell and Mrs. Clark drew marbles from a mystery jar of 100 marbles. Ms. Hammell drew 55 pink marbles and Mrs. Clark drew 70 pink marbles.  Find p hat, standard error, and margin of error for Ms. Hammell and Mrs. Clark.

A school is going to offer a new class next year. The school polls the students to see which class they would prefer. 47% respond that they would like East Asian History. 53% respond that they would like Southern African Studies. The margin of error for the poll is +4%. Explain whether the poll clearly indicates the majority's preference.

198 students wear pretty costumes, 93 students wear traditional costumes. About how many students plan to wear a pretty or traditional costume?

Fill in the remaining entries in the two-way table shown to the right. A survey of 140 patrons at a restaurant gave the preferences for entrees and drinks shown to the right. Fill in the missing entries in the table. Vegetarian Meat/Fish 51 Wine No Wine Total 38 Total 66 140

To determine the proportion of voters who favor a certain candidate for governor, the campaign staff phones 2500 residents of the state chosen from state property tax rolls. The 2500 residents represent what?