True or False (explain your answer). 1.True or false. If all the coefficients a1, a2, ..., an in the objective function P = a1x1 + a2x2 + ... + anxn are nonpositive, then the only solution of the problem is x1 = x2 = ... = xn and P = 0. 2. True or false. The pivot column of a simplex tableau identifies the variable whose value is to be decreased in order to increase the value of the objective function (or at least keep it unchanged). 3. True or false. The ratio associated with the pivot row tells us by how much the variable associated with the pivot column can be increased while the corresponding point still lies in the feasible set. 4. True or false. At any iteration of the Simplex procedure, if it is not possible to compute the ratios or the ratios are negative, then one can conclude that the linear programming problem has no solution. 5. True or false. If the last row to the left of the vertical line of the final simplex tableau has a zero in a column that is not a unit column, then the linear programming problem has infinitely many solutions. 6. True or false. Suppose you are given a linear programming problem satisfying the conditions: The objective function is to be minimized. All the variables involved are nonnegative, and Each linear constraint may be written so that the expression involving the variables is greater than or equal to a negative constant. Then the problem can be solved using the Simplex method to maximize the objective function P = C. 7. True or false. The objective function of the primal problem can attain an optimal value that is different from the optimal value attained by the dual problem. 1. TRUE; all the coefficients are nonpositive so if one of the \"xi\" becomes positive (so not equal to 0) the value of the objective function would decrease, so the only possibility to stay at the optimum is x1=x2...xn=0 then P=0. 2. TRUE; the interest of the pivot method is to find the variables that have to be decreased to maximize the objective function. At the end of the Simplex tableau the expression of the objective function is found with the nonbasic variables, all the nonbasic variable that have a negative sign have to be minimized to optimize the objective function (or at least keep it unchanged). 3. TRUE; the smallest ratios chosen during each pivoting step guarantee that the solution stays in the feasible region. For instance, at the initial condition conditions the pivot column is the x column and the ratios correspond to the intersection of the boundary line \"y=0\" with the boundary lines \"u=0, v=0 and w=0\" respectively (u,v,w are slack variables), in order to stay in the feasible set the variable associated with the pivot column can be increased until the corresponding ratio is reached. 4. TRUE; a linear programming problem will have no solution if the Simplex Method breaks down during a step, such as when all the ratios are negative or if the computation is impossible. 5. TRUE; a linear programming problem will have infinitely many solutions in the only case the last row to the left of the vertical line in the final Simplex Tableau contains a zero in the column, which is not a unit column, then the linear programming problem has infinitely many solutions. 6. FALSE; all the nonvariables involved are nonnegative. 7. TRUE; in primal problems there is always a direction or subspace of directions for moving which increases the objective function from every suboptimal point. When one moves in such direction, it is said that he/she is removing slack between the candidate solution or even on or more constraints. On the other hand, in dual cases, the dual vector usually multiplies the constraints which determine the primal constraints position. When we vary the dual vector in the dual problem, it's the same as revising the primal problem upper bounds