Q.If the feasible region for a LPP is _________, then the optimal value of the objective function $Z = ax + by$ may or may not exist.

The key idea is that when the feasible region is unbounded, the objective function may still have a finite optimum if the region is bounded in the direction of optimization — otherwise, the optimum does not exist.

Reasoning:

1. In a linear programming problem, an unbounded feasible region means the set of constraints does not enclose the region in all directions; it extends infinitely in at least one direction.
2. If the objective function $Z = ax + by$ is to be maximized and the region is unbounded in the direction of increasing $Z$, then $Z$ can increase without bound — no finite maximum exists. …

Why This Happens: The Concept of an Unbounded Feasible Region

In Linear Programming, the feasible region is the set of all points $(x, y)$ that satisfy all the constraints. When this region is bounded, it's a closed polygon — you can draw a circle around it. In that case, the maximum and minimum of any linear objective function $Z = ax + by$ are guaranteed to exist (by the Extreme Value Theorem), and they occur at one of the corner points.

But when the region is unbounded, it stretches out to infinity in at least one direction. Imagine a wedge that never closes — you can keep moving along it forever. Now, whether $Z$ has an optimal value depends on which way you're trying to go.

If the objective function increases as you move outward in the unbounded direction, then $Z$ can grow without bound — no maximum exists. If it decreases in that direction, the maximum might still be at a finite corner point. Similarly, the minimum might be at a corner, or it might not exist if the function keeps decreasing into infinity.

The key insight: unbounded region does not mean no optimum exists — it means the existence depends on the coefficients $a$ and $b$ in $Z = ax + by$.

Step-by-Step Reasoning

1. What "unbounded" means geometrically

   A feasible region is unbounded if it is not contained within any circle of finite radius. For example, the region $x \geq 0, y \geq 0$ is unbounded — it extends infinitely in the positive $x$ and $y$ directions. In such a region, you can find points with arbitrarily large $x$ or $y$ coordinates.

2. How the objective function behaves in an unbounded region

   Consider $Z = ax + by$. If the region is unbounded in a direction where $Z$ increases, then by moving further out, you can make $Z$ arbitrarily large — so no maximum exists. Conversely, if $Z$ decreases in that unbounded direction, the maximum might be at a finite corner point. The same logic applies to the minimum.

3. A concrete example where the maximum exists

   Suppose the feasible region is $x \geq 0, y \geq 0, x + y \geq 1$ (unbounded above and to the right). Let $Z = -x - y$. As $x$ and $y$ increase, $Z$ becomes more negative — so the maximum occurs at the smallest possible $x+y$, which is at the corner $(0,1)$ or $(1,0)$. Here, the maximum exists even though the region is unbounded.

4. A concrete example where the maximum does not exist

   Same region $x \geq 0, y \geq 0, x + y \geq 1$, but now $Z = x + y$. As you move outward, $Z$ increases without bound — no maximum exists. The minimum, however, exists at the corner points where $x+y=1$.

5. The general rule …

Method: Reading the "may or may not exist" clue

When a statement says the optimum may or may not exist and asks what kind of feasible region causes this, decide by contrast between the two region types.

The reasoning pattern

Step 1: Recall the guarantee for a bounded region.

If the feasible region is bounded (fits inside a circle), a continuous linear $Z = ax + by$ is guaranteed to attain both a maximum and a minimum, each at a corner. So existence is certain -- this cannot be the case the blank describes.

Step 2: Recall what an unbounded region allows. …

Common Mistakes

Mistake 1: Answering "bounded" instead of "unbounded."

Why it's wrong: a bounded region guarantees both a maximum and a minimum, so it can never produce the "may or may not exist" behaviour the sentence describes. That uncertainty is exactly what an unbounded regio …