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Question #40526

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [0, 3] \to [0, 4]$ be a function defined by $f(x) = x^4 - 6x^3 + 13x^2 - 12x + 4$ . Let $g(x)$ denote the inverse function of $f(x)$ . Determine the value of $g'(4)$ , if it exists.

Options

A.
$\frac{1}{12}$
B.
$-\frac{1}{12}$
C.
$\frac{1}{4}$
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

Even Multiplicity Roots|Injective Mapping (One-One)|Bijectivity and Inverse

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to verify the prerequisites of the Inverse Function Theorem, specifically domain restriction and injectivity.

The polynomial is designed with even multiplicity roots at $x=1$ and $x=2$ , which create local minima and destroy monotonicity on the given interval $[0, 3]$ .

A common trap is for students to mechanically solve $f(x) = 4$ (finding $x=3$ ), and blindly apply the formula $g'(y) = 1/f'(x)$ to get $1/12$ , completely ignoring that $f(0)=4$ as well.

By failing to check if the function is globally injective on the specified domain, they attempt to evaluate the derivative of an inverse function that does not mathematically exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free