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Question #40965

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3 - 3x^2 + 3x$ . Let $g(x)$ be the inverse function of $f(x)$ . Consider the integral $I = \int_{0}^{2} \frac{(g''(x))^2}{(g'(x))^3} \, dx$ Which of the following is the correct value of $I$ ?

Options

A.
-8
B.
8
C.
0
D.
$\infty$

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

Polynomial Continuity/Differentiability Link, Mean Value Theorem Applicability, The Sign-Consistency Principle, Inverse Function Integration

Why This Question Is Tricky

Common Mistakes

This question tests the applicability of the Fundamental Theorem of Calculus (FTC) when integrating expressions involving the derivatives of an inverse function.

The trap lies in the Sign-Consistency Principle and the behavior of the polynomial's derivative.

Students typically use the substitution $x = f(t)$ to simplify the integrand, arriving at $\int_0^2 4/(t-1)^2 dt$ .

The lethal mistake is blindly applying the FTC across the critical point $t=1$ where $f'(t)$ vanishes, yielding a negative value ( $-8$ ) for a strictly positive integrand.

A global analysis of $f'(t)$ reveals a non-integrable singularity at $t=1$ , meaning the integral actually diverges to $\infty$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Polynomial Continuity/Differentiability Link
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free