State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective function $Z = ax + by$ may or may not exist.

State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum

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Z = ax + by

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Concept understanding — Unbounded Feasible Region

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.

Note

Unbounded does not mean "huge area" — it means the region has no boundary in some direction, so it stretches off to infinity there.

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.

Watch out

An unbounded region does not automatically destroy the answer. For $Z = -x + y$ on the first quadrant, as $x \to \infty$ the term $-x$ drives $Z \to -\infty$ , so a maximum can still sit at a corner. Always compare the objective's direction with the open direction of the region.

Exam technique

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

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True. An unbounded feasible region does not by itself decide the question — whether $Z = ax + by$ attains a maximum or a minimum depends on the direction of the objective relative to the open direction of the region.

What "unbounded" means

A feasible region is unbounded when it stretches to infinity in at least one direction — you can keep moving inside it without ever crossing a boundary.

Why the optimum may or may not exist

Evaluate $Z$ at the corner points to get a candidate value $M$ , then, because the region is open in some direction, run the extra graphical check:

For a maximum: if the half-plane $ax + by > M$ still meets the region, $Z$ can be pushed higher without end, so no maximum exists; if it misses the region, $M$ is the maximum.
For a minimum: do the same test with $ax + by < M$ .

Both outcomes are genuinely possible: …

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Q.State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective function $Z = ax + by$ may or may not exist.

Concept understanding — Unbounded Feasible Region

Unbounded Feasible Region

How a region becomes unbounded

Why it matters for the objective

Exam technique

What "unbounded" means

Why the optimum may or may not exist

Method: Judging a True/False claim about unbounded optima

The reasoning pattern

Common Mistakes

Q.State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective function Z=ax+byZ = ax + byZ=ax+by may or may not exist.

What "unbounded" means

Why the optimum may or may not exist

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Q.State whether the following statement is True or False: If the feasible region for a LPP is unbounded, maximum or minimum of the objective function $Z = ax + by$ may or may not exist.

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