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Question #41696

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in the first quadrant bounded by the $y$ -axis, the line $y = 2$ , and the curve $y = \int_0^x |t^2 - 1| dt$ . If $A$ is the area of the region $R$ , find the maximum value of the objective function $Z = 4A x - 5y$ over the region $R$ .

Options

A.
$10\sqrt{3} - \frac{20}{3}$
B.
$10$
C.
$12\sqrt{3} - 8$
D.
$\frac{25}{2}$

Check Your Answer

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

Corner Point Method Precision|Inequalities in Definite Integrals|The Sign-Consistency Principle|Horizontal vs Vertical Strips

Why This Question Is Tricky

Common Mistakes

This question tests 'Corner Point Method Precision' and the 'Sign-Consistency Principle'.

Students often blindly apply the corner point method, assuming the maximum of a linear objective function over a bounded region occurs at a vertex (yielding 10 at the corner point $(2,2)$ ).

However, because the feasible region has a curved boundary, the true maximum can occur where the objective function's level curve is tangent to the boundary.

Additionally, calculating the area $A$ acts as an efficiency filter: using vertical strips requires tedious piecewise integration, whereas recognizing the horizontal strip equivalent $A = \int x \, dy$ reduces it to a simple integral, provided the student correctly accounts for the modulus sign-change at $x=1$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Corner Point Method Precision
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free