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Question #41755

Difficulty: 8/10 | 🟢 Free

The Question

A variable plane intercepts the coordinate axes at the points $A(a, 0, 0)$ , $B(0, b, 0)$ , and $C(0, 0, c)$ . It is given that $a + b + c = 2$ and $|a| \le 2, |b| \le 2, |c| \le 2$ . If $O$ is the origin, then the maximum possible volume of the tetrahedron $OABC$ is

Options

A.
$\frac{4}{3}$
B.
$\frac{4}{81}$
C.
$\frac{8}{27}$
D.
$\frac{2}{3}$

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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

Variable plane intercepts|Arithmetic Mean Inequality|Modulus Inequality Logic|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the domain of validity for the AM-GM inequality and analyze global extrema on restricted intervals.

The common trap is blindly applying AM-GM to $a+b+c=2$ (assuming positive intercepts), which yields a local maximum volume of $4/81$ .

However, the modulus constraint $|a|, |b|, |c| \le 2$ permits negative intercepts.

By assigning one variable a negative value, the product's magnitude can be significantly increased, forcing students to construct a single-variable function and evaluate it at the boundary of the domain to find the true global maximum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Variable plane intercepts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free