4x3 + x2 Task 1 (50 points) Consider the following LP problem: maximize 3x1 + 2x2 + s.t. X1 + 2x3 4 2x1 + 3x3 5 5 2xi + x2 + 3x3 5 7 X1, X2X3, X4 > 0 and call x4, X5, X6 the three slack variables to bring the problem in standard form. Subtask 1.a Using only matrix computations, like in the revised simplex method, calculate the value of the variables corresponding to a solution with XB = (x1, x2, X6) basic variables and xy = (x3, x4, X5) non basic variables. [You can use any program of your choice to invert the matrix Ag] Subtask 1.b Show that the solution found at the previous subtask is optimal Subtask 1.c Which is the value of the variables in the optimal solution of the dual problem? Which constraints are "binding and which have slack in this optimal solution? Subtask 1.d If the second constraint's right-hand side is changed to 7 would the current basic solution remain feasible and optimal and would the objective function increase? Subtask 1.e If the objective function is changed to 3x1 + 2x2 + 6x3 would the basic structure change? If so which variable would enter the basis and which would leave? Subtask 1.1 A new variable is introduced with coefficients of 3,4,6 in the first second, and third con- straint, respectively. Determine what cost coefficient should the new variable have in order to have a change in the structure of the optimal basic solution