Before You Begin

Question #41914

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} x^2 & |x-1| & 3 \\ x & x-1 & 1 \\ 1 & 0 & 1 \end{vmatrix}$ . Consider the application of Rolle's Theorem for the function $f(x)$ on the interval $[-1, 3]$ . The value of $c \in (-1, 3)$ such that $f'(c) = 0$ is:

Options

A.
$\frac{2 + \sqrt{7}}{3}$
B.
$\frac{2 - \sqrt{7}}{3}$
C.
$\sqrt{\frac{5}{3}}$
D.
1

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

Mean Value Theorem Applicability|Differentiation of Determinants|Modulus Sign Dependency|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

The problem tests the applicability of Rolle's Theorem on a determinant containing a modulus function.

Students often blindly differentiate the determinant or ignore the modulus sign dependency, leading to incorrect critical points from the wrong piecewise branch (e.g., $(2-\sqrt{7})/3$ or $\sqrt{5/3}$ ).

The trap forces them to perform a 'Modulus Function Breakdown' to ensure global differentiability and select the valid root strictly within the correct domain interval.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Mean Value Theorem Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free