Before You Begin

Question #41677

Difficulty: 8/10 | 🟢 Free

The Question

A factory produces two liquids, $X$ and $Y$ . The daily production constraints are $x + y \ge 6$ , $x \le 6$ , and $y \le 6$ , where $x, y \ge 0$ are the quantities in kiloliters. The daily operating cost (in thousands of rupees) is modeled by the objective function $C(x,y) = 4|x - y| + x + y$ . What is the minimum possible daily operating cost?

Options

A.
6
B.
12
C.
18
D.
30

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Objective Function Gradient|Word Problem Modeling|Sign Flipping Rule|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the 'Sign Flipping Rule' and modulus dependency in Linear Programming Problems.

Students typically evaluate the objective function only at the standard vertices of the feasible region polygon (here, $(6,0), (0,6), (6,6)$ ), which incorrectly yields a minimum of 12.

However, the presence of the modulus $|x - y|$ introduces a non-differentiable ridge along the line $x = y$ .

The gradient of the objective function abruptly changes direction across this line.

Because the gradients in both sub-regions point towards $x = y$ , the optimal solution is forced onto this internal boundary, creating a new critical point at $(3,3)$ that is not a vertex of the original constraints.

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