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Question #61056

Difficulty: 8/10 | 🟢 Free

The Question

Let $a, b, c$ be real constants such that $c > b > 0$ and $a > 1$ . A function $f(x)$ is defined as $f(x) = \prod_{n=1}^{\infty} (x+a)^{\left(\frac{x}{b}\right)^n}$ Which of the following represents $f'(c)$ ?

Options

A.
$f(c) \left[ \frac{b \ln(c+a)}{(b-c)^2} + \frac{c}{(b-c)(c+a)} \right]$
B.
$f(c) \left[ \frac{c \ln(c+a)}{(b-c)^2} + \frac{b}{(b-c)(c+a)} \right]$
C.
$f(c) \left[ \frac{b \ln(c+a)}{(b-c)^2} - \frac{c}{(b-c)(c+a)} \right]$
D.
\text{Does not exist}

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

Logarithmic Differentiation|Selection of GP Terms|Infinite GP Convergence

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's rigor in checking the domain of convergence for infinite series before applying calculus operations.

A common trap in JEE Advanced is presenting an infinite product or series that can be simplified using standard formulas (like the sum of an infinite GP), tempting students to immediately apply logarithmic differentiation.

However, the sum formula $S = a/(1-r)$ is only valid for $|r| < 1$ .

By asking for the derivative at a point $x = c$ where the common ratio exceeds 1, the question punishes algorithmic execution that lacks conceptual verification.

Students who ignore the convergence condition will incorrectly select Option A.

Quick Facts About This Question

Question Information

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