Suppose that an optimal solution to the problem: minimize ex subject to ax > b, x > 0, exists. prove the following complementarity theorem. a. a variable is zero for all primal optimal solutions if and only if its complementary dual variable is positive for some dual optimal solution. b. a variable is unbounded in the primal feasible set if and only if its complementary dual variable is bounded in the dual feasible set. c. a variable is unbounded in the primal optimal set if and only if its complementary dual variable is zero for all dual feasible solutions. d. a variable is positive for some primal feasible solution if and only if its complementary dual variable is bounded in the dual optimal set.

Robots with telescoping arms are sometimes used to perform tasks (e. g., welding or placing screws) where access may be difficult for other robotic types. during a test run, a robot arm is programmed to extend according to the relationship r = 3 + 0.5cos(4θ) and the arm rotates according to the relationship θ=−π4t2+πt , where r is in feet, θ is in radians, and t is in seconds. use a computer program to plot the path of tip a in x and y coordinates for 0 ≤ t ≤ 4s.

Why would a programmer use the logical operator and in an if statement? a: when an action is to be taken that requires both conditions to be falseb: when an action is to be taken that requires both conditions to be truec: when an action is to be taken that requires the first condition to be falsed: when an action is to be taken that requires the second condition to be truei took the test and the answer is b.

Jaina and tomas are being considered as new tenants in an apartment. the landlord looks at their creditworthiness because he wants to be sure his new tenant pays the rent on time and in full. the table below summarizes the information that was on their applications. application information questions jaina tomas how many years have you had your job? 5 2 what is your monthly salary? $1,850 $2,500 how many credit cards do you have? 4 1 how much debt do you have? $13,000 $7,000 how many times were you late with payments on credit cards in the past year? 5 1 who will the landlord decide to be more creditworthy and why? tomas because the ratio of his debt to income is less. jaina because she has had her job longer, which makes her look more stable. jaina because she has more credit cards available to her. tomas because he makes more money per month.