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Question #41771

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \left\{ (x, y) \in \mathbb{R}^2 : 4x^2 + 9y^2 \le 36 \text{ and } 3y > 2x \right\}$ . Find the maximum value of the function $f(x, y) = 2x + 3y$ on the set $S$ .

Options

A.
$6\sqrt{2}$
B.
12
C.
6
D.
\text{Does not exist}

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

Objective Function Gradient|Tangent/Normal Slope Interplay|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to optimize a linear objective function over a region with non-linear constraints while navigating the topological trap of open boundaries.

Students typically use parametric coordinates or Lagrange multipliers to find the critical point where the gradient of the objective function aligns with the normal of the ellipse.

The trap lies in the Weierstrass Extreme Value Theorem: because the constraint $3y > 2x$ is a strict inequality, the domain is not compact.

The unique global maximum of the objective function on the closed ellipse lies exactly on this excluded boundary.

Consequently, the function approaches a supremum but never actually attains it within the set $S$ , meaning the maximum does not exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Objective Function Gradient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free