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Question #40544

Difficulty: 8/10 | 🟢 Free

The Question

Let $g(x) = |x^2 - 4x + 3|$ . Find the solution set $S$ of the inequality $\frac{g'(x)}{[x] - 2} \le 0$ in the domain $x \in [-2, 4]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $g'(x)$ is the derivative of $g(x)$ .

Options

A.
$(1, 2)$
B.
$[-2, 1) \cup (2, 3)$
C.
$(1, 2) \cup (2, 3)$
D.
$[1, 2)$

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

Cross Multiplication Trap|Step Function Discontinuities|Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question exploits the 'Cross-Multiplication Trap' and 'Chain Rule Inner Function Differentiability'.

Students commonly cross-multiply the denominator $[x] - 2$ , incorrectly solving $g'(x) \le 0$ to get $[-2, 1) \cup (2, 3)$ .

They frequently miss that $[x] - 2 = 0$ for the entire interval $[2, 3)$ , not just at $x=2$ , which invalidates any solution in that range (a trap leading to option C).

Additionally, the absolute value function makes $g'(x)$ undefined at the roots $x=1$ and $x=3$ .

Failing to exclude these points of non-differentiability traps students into selecting closed intervals.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Cross Multiplication Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free