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Question #41325

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3, z_4$ be the roots of the equation $z^4 - 2z^3 + 3z^2 - 4z + 5 = 0$ . Let $A(x)$ be a $4 \times 4$ matrix whose entries are given by $A_{ij} = z_i z_j$ for $i \neq j$ and $A_{ii} = \sqrt{|x^3 - 3x + 2|} + z_i^2$ for $1 \le i, j \le 4$ . If $f(x) = \det(A(x))$ , find the number of points $x \in \mathbb{R}$ where $f(x)$ is NOT twice differentiable.

Options

A.
1
B.
2
C.
3
D.
0

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

Second Order Derivative Continuity|Differentiation of Determinants|Vieta’s Formulas in Complex Domain|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of linear algebra (eigenvalues of rank-1 updates), theory of equations (Vieta's formulas), and advanced calculus (differentiability of absolute value functions).

The primary trap lies in the domain restrictions and singular points of the second derivative.

Students often correctly identify the roots of the inner polynomial $u(x) = x^3 - 3x + 2$ as $x=1$ and $x=-2$ , and mistakenly assume the second derivative fails to exist at both points due to the $|u(x)|^{3/2}$ term.

However, they must analyze the local behavior: at the repeated root $x=1$ , the derivative $u'(1)=0$ suppresses the singularity, making the function twice differentiable.

The breakdown of continuity and differentiability only occurs at the simple root $x=-2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Second Order Derivative Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free