For each of the following linear programs: (1) Sketch the feasible region of the linear program and the direction of the objec- tive function. (2) Use you sketch to find an optimal solution to the program. State the optimal solution and give the objective value for this solution. If an optimal solution does not exist, state why. (a) (b) - 4x₁ + x₂ subject to x₁ + x₂ ≤ 2, x1 - 2x₂ ≤ 4, x₁ + x₂ ≥ 7, X1, X₂0 maximize maximize subject to ₁ + 2x₂ 1x1 -₁ + 2x₂ ≤ 4, 1+ 3x2 ≤ 12, X1, X20

Use the simplex method to solve the linear programming problem. Maximize: z = 2x₁ + x2 subject to: x₁ + 4x₂ ≤ 12 2x₁ + 6x₂ ≤ 4 X₁ + 2x2 ≤ 4 with x₁ ≥ 0, x₂ ≥ 0. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The maximum is when x₁ = x₂ = , S₁ = S2 = and S3 = B. There is no maximum solution to this linear programming problem.