Q.The feasible region of a linear programming problem is an unbounded region lying between two parallel straight lines that are both parallel to the line $3x - 4y = 0$. The lower boundary is the line $3x - 4y = -16$, which passes through the point $(0, 4)$, and the upper boundary is the line $3x - 4y = 12$, which passes through the point $(12, 6)$; the region consists of all points with $x \ge 0$, $y \ge 0$ satisfying $-16 \le 3x - 4y \le 12$, and its relevant corner points are $(0, 4)$ and $(12, 6)$. Let $F = 3x - 4y$ be the objective function. The maximum value of $F$ is
(A) $0$
(B) $8$
(C) $12$
(D) $-18$

Unbounded Feasible Region

Stand in an open field and you can walk as far as you like in some direction without ever hitting a wall. Stand in a closed hall and walls stop you on every side. A feasible region in linear programming is the set of all points satisfying every constraint; calling it unbounded means that in at least one direction you can keep moving forever and still stay inside it.

How a region becomes unbounded

Each constraint is a linear inequality such as $x \geq 0$, $y \geq 0$, or $x + y \leq 10$, and each cuts the plane into a half-plane. The feasible region is the overlap of these half-planes. If the constraints fail to "close off" the region in some direction, it runs on without end.

Example. With only $x \geq 0$ and $y \geq 0$, the region is the entire first quadrant — you can travel as far as you wish along either axis, so it is unbounded. Add $x + y \leq 10$ and the region shrinks to the triangle with vertices $(0,0)$, $(10,0)$, $(0,10)$ — now bounded.

Why it matters for the objective

In an LP problem you maximise or minimise $Z = ax + by$ over the feasible region.

• If the region is bounded, both the maximum and the minimum of $Z$ exist and each occurs at a corner point.
• If the region is unbounded, an optimum may fail to exist — you might push $Z$ larger and larger by moving out along the open direction.

Exam technique

1. Graph the constraints and evaluate $Z$ at every corner point. …