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Question #41665

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in the first quadrant of the $xy$ -plane defined by the system of inequalities: $x \le 5$ $y \le 6$ $x^2 y - 8xy + 16y \le x^3 - 8x^2 + 16x$ Find the maximum value of the objective function $Z = 4x + 5y$ over the region $R$ .

Options

A.
45
B.
46
C.
50
D.
54

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Boundary Value Theorem|Global Extrema on Intervals|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

The problem tests the 'Zero-Factor Cancellation Trap' within a Linear Programming context.

Students often divide both sides of the constraint inequality by the common factor $(x-4)^2$ without considering the case where it equals zero.

This erroneous cancellation reduces the constraint strictly to $y \le x$ , causing the student to miss the valid line segment at $x=4$ where $y$ can extend up to 6.

Consequently, they optimize the objective function over an incomplete feasible region, yielding the sub-optimal trap answer of 45 instead of the true global maximum of 46.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free