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Question #40617

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [1, \infty) \to (-\infty, \frac{1}{6}]$ be defined as $f(x) = \sum_{k=0}^{5} \binom{5}{k} \frac{(-1)^k}{k+1} x^{k+1}$ Let $g(x)$ be the inverse function of $f(x)$ . Find the value of $14 \int_{-21/2}^{0} g(x) dx$

Options

A.
401
B.
-107
C.
481
D.
40

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

General Term Formulation|Inverse Function Integration|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to condense a binomial series into a closed-form function and integrate its inverse using integration by parts.

The primary trap lies in the domain restriction: $f(x)$ is not globally injective, as $f'(x) = (1-x)^5$ changes sign at $x=1$ .

The equation $f(x) = c$ has multiple roots (e.g., $f(x)=0$ at $x=0$ and $x=2$ ).

Students who ignore the specified domain $[1, \infty)$ will incorrectly choose the branch $(-\infty, 1]$ , leading to the wrong limits for the inverse integral ( $y \in [-1, 0]$ instead of $y \in [3, 2]$ ) and arriving at the trap answer -107.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	General Term Formulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free