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Question #40954

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \log_{([x] + \{x\}^3)} (x^3)$ for $x > 1$ , where $[x]$ and $\{x\}$ denote the greatest integer and fractional part functions respectively. If $L$ and $R$ are the left-hand and right-hand derivatives of $f(x)$ at $x=2$ , then the value of $R - L$ is:

Options

A.
$\frac{9}{2 \ln 2}$
B.
$\frac{3}{2 \ln 2}$
C.
0
D.
$\frac{3}{\ln 2}$

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

Chain Rule Complexity|Logarithmic Differentiation Domain|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the application of the chain rule and logarithmic differentiation on a composite function containing piecewise components.

The core trap lies in treating the Greatest Integer Function (GIF) and fractional part as globally differentiable entities.

Students often blindly apply the chain rule, assuming $\frac{d}{dx}\{x\} = 1$ everywhere, or they evaluate $\{x\} = 0$ at $x=2$ before taking the left-hand limit, incorrectly concluding that the derivative of the base is $0$ from both sides.

This leads to the erroneous conclusion that $L = R$ , yielding an answer of $0$ .

The problem rigorously tests the proper evaluation of one-sided derivatives for piecewise functions at critical integer boundaries.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Chain Rule Complexity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free