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Question #41674

Difficulty: 8/10 | 🟢 Free

The Question

Consider the linear programming problem: Maximize $Z = \left( \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} \right) x + \left( \lim_{x \to 0} \frac{\ln(\cos(\sqrt{2}x))}{x^2} \right) y$ subject to the constraints: $x - 2y \le \lim_{x \to 0} \frac{e^{x^2} - \cos(\sqrt{2}x)}{x^2}$ $x + y \ge \lim_{x \to 0} \frac{e^{3x} - 1}{\sin x}$ $x \ge 0, y \ge 0$ Which of the following statements is true regarding the optimal solution?

Options

A.
The maximum value of $Z$ is $1$ , and it occurs at exactly one point.
B.
The maximum value of $Z$ is $1$ , and it occurs at infinitely many points.
C.
The maximum value of $Z$ does not exist because the feasible region is unbounded.
D.
The maximum value of $Z$ is $\frac{3}{2}$ , occurring at $(3,0)$ .

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Unbounded Region Trap|Multiple Optimal Solutions|The L'Hôpital's Rule Misuse|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the 'Unbounded Region Trap' combined with 'Multiple Optimal Solutions' in a Linear Programming Problem masked by calculus limits.

Students must first correctly evaluate indeterminate forms using L'Hôpital's Rule or Taylor expansions to reveal the true LPP constraints.

The primary trap lies in the feasible region being unbounded; a common misconception is that an unbounded region implies the maximum value of the objective function does not exist.

However, because the objective function $Z = \frac{1}{2}(x - 2y)$ is parallel to the bounding constraint $x - 2y \le 2$ , the maximum value is strictly bounded by the constraint itself.

Furthermore, since this boundary extends infinitely within the feasible region, the maximum is achieved at infinitely many points, punishing students who only check finite vertices or blindly assume unboundedness precludes a maximum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Unbounded Region Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free