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Question #41669

Difficulty: 8/10 | 🟢 Free

The Question

Let $x, y,$ and $z$ be the side lengths of a non-degenerate triangle such that $z = 1$ and the perimeter of the triangle is $3$ . Consider the function $f(x, y) = x^3 + y^3 - 3xy$ . If $M$ and $m$ are the supremum and infimum of $f(x, y)$ respectively, then the value of $4(M - m)$ is:

Options

A.
9
B.
36
C.
27
D.
18

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

Boundary Value Theorem|LMVT Tangent Parallelism|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between optimization (Lagrange Multipliers/LMVT Tangent Parallelism) and geometric domain constraints.

Students typically find the interior critical point using gradients but often fail to correctly identify the boundary of the domain.

The classic trap is assuming the variables only need to be positive ( $x, y > 0$ ), leading to the incorrect boundaries $x \in (0, 2)$ and a false supremum of $8$ (yielding an incorrect final answer of $36$ ).

The 'Triangle Inequality Mind Game' forces the realization that the geometric constraints $x+z>y$ and $y+z>x$ strictly bound the domain to $x \in (1/2, 3/2)$ , drastically altering the supremum and testing the student's topological awareness.

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