Before You Begin

Question #40524

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x, y) \in \mathbb{R}^2$ satisfying the following constraints: $(y - x)^2 \le (x-1)^2 (x-6)^2 (2-x)$ $x \ge 1$ Find the maximum value of the objective function $Z = 3x + 4y$ for $(x, y) \in S$ .

Options

A.
14
B.
24
C.
30
D.
42

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

Even Multiplicity Roots|Boundary Value Theorem|Feasible Region Constancy

Why This Question Is Tricky

Common Mistakes

This question tests the 'Feasible Region Constancy' trap via even multiplicity roots.

Students typically solve the domain inequality $(x-1)^2(x-6)^2(2-x) \ge 0$ by dividing out the squared terms, incorrectly concluding the domain is strictly $x \in [1, 2]$ .

They fail to realize that the even multiplicity root at $x=6$ satisfies the non-negativity constraint without crossing the axis, creating a degenerate, isolated point $(6,6)$ in the feasible region.

By bounding the continuous portion of the region (which yields a maximum Z < 30), the problem ensures the absolute maximum occurs at this easily overlooked isolated point (Z=42), punishing blind algebraic simplification and rewarding rigorous topological analysis of the domain.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free