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Question #40505

Difficulty: 8/10 | 🟢 Free

The Question

Let $x, y, z$ be distinct positive real numbers such that $xyz = 8$ . Let $M$ be the minimum value of the quadratic expression $f(t) = (x+y+z)t^2 + 2(x+y+z)t + \frac{(x+y+z)^2 + 54}{x+y+z}$ over all $t \in \mathbb{R}$ . Find the maximum possible value of $[M]$ , where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
7
B.
8
C.
9
D.
10

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

Maximum/Minimum of Quadratic Expressions|Arithmetic Mean Inequality|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of quadratic optimization and the AM-GM inequality, specifically targeting the boundary conditions of the Greatest Integer Function (GIF).

The 'Lethal Collision' occurs because the variables are explicitly stated to be 'distinct'.

Students commonly apply AM-GM to find $x+y+z \ge 6$ , leading to a minimum value $M \le 9$ , and incorrectly conclude the maximum of $[M]$ is 9.

However, the distinctness of the variables forces a strict inequality $x+y+z > 6$ , meaning $M$ approaches 9 but never reaches it.

Thus, the true maximum of $[M]$ is 8, catching students in a classic off-by-one error.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Maximum/Minimum of Quadratic Expressions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free