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Question #40957

Difficulty: 8/10 | 🟢 Free

The Question

Let $a_0, a_1, a_2, \dots, a_{402}$ be real numbers such that $\sum_{r=0}^{402} a_r x^r = (1 - 4x + x^2)^{201}$ for all $x \in \mathbb{R}$ . Let $f: D \to \mathbb{R}$ be defined by $f(x) = \ln \left( \sum_{r=0}^{402} a_r x^r \right)$ where $D \subseteq \mathbb{R}$ is the maximal domain of $f$ . Find the value of $f'(1)$ .

Options

A.
201
B.
-201
C.
0
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|Multinomial Theorem|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests a student's vigilance regarding domain restrictions when applying logarithmic differentiation to multinomial expansions.

The trap is set by defining a function as the natural logarithm of a polynomial expanded via the Multinomial Theorem.

Students who blindly apply the chain rule $f'(x) = P'(x)/P(x)$ will calculate a neat integer value (201) at $x=1$ .

However, evaluating the base polynomial at $x=1$ yields a negative number.

Since the argument of a real-valued logarithm must be strictly positive, $x=1$ lies outside the function's domain, making the derivative undefined.

This catches students who focus purely on derivative mechanics while ignoring fundamental domain constraints.

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