Consider maximizing $P = x + y$ subject to $x > 0$, $y > 0$, and $x + y < 10$. Why does this linear programming problem have no maximum value?

The feasible region is unbounded.

The optimal value occurs on a boundary that is excluded from the feasible region.

The objective function is not linear.

The constraints are contradictory and define an empty region.

Consider maximizing $P = 5x - y$ subject to $x \leq 6$, $y \geq 1$, and $x + y < 10$. Does an optimal maximum value exist for $P$?

Yes, because the optimal vertex (6, 1) is included in the feasible region.

No, because the inequality $x + y < 10$ is strict.

No, because the feasible region is unbounded.

Yes, but the optimal value occurs on the line $x + y = 10$.