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Question #41342

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [1, e] \to \mathbb{R}$ be defined by $f(x) = \int_1^x t \ln t \, dt + \int_1^x [t] \, dt$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . If $g(x)$ is the inverse function of $f(x)$ , then the value of $g'\left(2 \ln 2 + \frac{1}{4}\right)$ is:

Options

A.
$\frac{1}{2 \ln 2 + 2}$
B.
$\frac{1}{2 \ln 2 + 1}$
C.
\text{Does not exist}
D.
$\frac{1}{2 \ln 2}$

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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

Inverse Function Derivative|Integration by Parts (LIATE Rule)|Integral Part and Fractional Part|Logarithmic Differentiation Domain

Why This Question Is Tricky

Common Mistakes

This question tests the differentiability of inverse functions combined with piecewise continuous integrands.

The trap lies in blindly applying the inverse derivative formula $g'(y) = 1/f'(x)$ and substituting $x=2$ into the formal derivative $f'(x) = x \ln x + [x]$ , which yields $2 \ln 2 + 2$ (Option A).

However, because the integrand $[t]$ has a jump discontinuity at $t=2$ , $f(x)$ has a corner at $x=2$ (different left and right derivatives).

Consequently, the inverse function $g(x)$ also has a corner at the corresponding $y$ -value, making it non-differentiable there.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Inverse Function Derivative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free