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Question #40527

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \lim_{y \to 0} \left( \frac{x^2 - 5x + 4 + y}{x^2 - 5x + 4} \right)^{\frac{[x]}{y}}$ for $x \in (1, 4)$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Find the number of points of local extrema of $f(x)$ in the interval $(1, 4)$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Denominator Restriction|The 1^∞ Form Shortcut|Greatest Integer Function Traps|First Derivative Test Trap

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of $1^\infty$ limits combined with piecewise function analysis.

The primary trap lies in the First Derivative Test.

Students often differentiate the exponent ignoring the Greatest Integer Function (GIF), incorrectly finding a single maximum at $x=2$ .

Alternatively, they find the true critical point at $x=2.5$ but fail to check the non-differentiable integer boundaries.

Because the GIF creates jump discontinuities, the First Derivative Test is insufficient.

At $x=2$ , the function jumps down from $-0.5$ to $-1$ and then increases, creating a hidden local minimum that traps students who assume continuity.

Quick Facts About This Question

Question Information

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