The feasible region is the area where all constraints are satisfied.
A bounded feasible region is enclosed and does not extend to infinity in any direction.
The objective function z=5x−9y represents what needs to be maximized or minimized.
The Fundamental Theorem of Linear Programming states:
If the feasible region is bounded, then the objective function always attains both maximum and minimum values at the corner points (vertices) of the feasible region.
Evaluating the options:
Option 1 (only maximum): Incorrect. Bounded regions provide both maximum and minimum values.
Option 2 (only minimum): Incorrect. Bounded regions provide both maximum and minimum values.
Option 3 (both maximum and minimum): Correct. This follows directly from the fundamental theorem.
Option 4 (neither maximum nor minimum): Incorrect. This only occurs when the region is unbounded in certain directions.