Consider the following problem Maximize z = 3x1 + x2 + 4x, subject to 6x1 + 3x2 + 5x3 S 25 3x + 4x2 + 5x, s 20 X1, X2, X3 20 After applying the simplex method, the optimal tableau is as shown below. Basic Variable Eq. No. Coefficient of X2 3 Right Side x Si S2 Z 0 1 0 2 0 17 1 0 1 1 0 5 3 0 1 1 3 2 X3 2 1 5 0 (c) Write down the optimal solution for the dual problem from the optimal tableau and verify this solution by solving the dual problem graphically. () Suppose that the original problem is changed to Maximize z = 3x1 + 3x2 + 4x3 subject to 6x2 + 2x2 + 5x3 S 25 3x + 3x2 + 5x3 s 20 X1, X2, X3 20 Use duality to determine whether the previous solution is still optimal. (e) Now suppose the only change in the original problem is that a new variable x.. has been introduced into the model as following: Maximize z = 3x + x2 + 4x3 + 2x new subject to 6x2 + 3x2 + 5x2 + 3xnews 25 3x1 + 4x2 + 5x3 + 2xnews 20 X1, X2, X3, Xnew 20 Use duality to determine whether the previous solution, along with Xnew = 0, is still optimal. Consider the following problem Maximize z = 3x1 + x2 + 4x, subject to 6x1 + 3x2 + 5x3 S 25 3x + 4x2 + 5x, s 20 X1, X2, X3 20 After applying the simplex method, the optimal tableau is as shown below. Basic Variable Eq. No. Coefficient of X2 3 Right Side x Si S2 Z 0 1 0 2 0 17 1 0 1 1 0 5 3 0 1 1 3 2 X3 2 1 5 0 (c) Write down the optimal solution for the dual problem from the optimal tableau and verify this solution by solving the dual problem graphically. () Suppose that the original problem is changed to Maximize z = 3x1 + 3x2 + 4x3 subject to 6x2 + 2x2 + 5x3 S 25 3x + 3x2 + 5x3 s 20 X1, X2, X3 20 Use duality to determine whether the previous solution is still optimal. (e) Now suppose the only change in the original problem is that a new variable x.. has been introduced into the model as following: Maximize z = 3x + x2 + 4x3 + 2x new subject to 6x2 + 3x2 + 5x2 + 3xnews 25 3x1 + 4x2 + 5x3 + 2xnews 20 X1, X2, X3, Xnew 20 Use duality to determine whether the previous solution, along with Xnew = 0, is still optimal