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Question #41317

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x & 0 & 2 \\ 0 & x & 0 \\ 2 & 0 & x \end{pmatrix}$ . Let $f(x) = (\det(A(x)))^{\det(A(x))}$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$ . Find the number of distinct real solutions to the equation $f'(x) = 0$ .

Options

A.
3
B.
4
C.
5
D.
6

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

Logarithmic Differentiation Domain|Matrix Trace Properties|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of domain constraints during symbolic differentiation.

Students correctly apply Jacobi's formula for the derivative of a determinant and log-differentiation to find the critical points.

However, a lethal trap awaits: they often fail to establish the domain of $f(x)$ , which requires the base of the exponential function to be strictly positive ( $\det(A(x)) > 0$ ).

By blindly solving $\text{tr}(\text{adj}(A(x))) = 0$ , they include the extraneous root $x = 2/\sqrt{3}$ , which yields a negative determinant, violating the log existence condition and leading to an incorrect count of 5 solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Logarithmic Differentiation Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free