In the solution region (x+y\leq 6), (2x+y\leq 8), (x\geq 0), and (y\geq 0), what is the maximum value of (2x+3y)?

Correct answer: (18)

The direct answer is option D, 18. The feasible region is in the first quadrant and is bounded by x+y≤6 and 2x+y≤8. Its vertices are (0,0), (4,0), the intersection (2,4), and (0,6). At these points, 2x+3y equals 0, 8, 16, and 18 respectively. Therefore the maximum is 18 at (0,6). Option A, 12, is not the largest value; it can be exceeded by the feasible point (0,6). Option B, 14, is likewise smaller than 18. Option C, 16, is the value at (2,4), but another vertex gives 18. Option D, 18, is correct. The corner-point method works because a linear objective over a closed polygon reaches its extreme value at a corner. Do not choose a value merely because it resembles a boundary number; calculate the objective at every vertex.