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Question #41264

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} 1 & \sin x & x + \cos x \\ -\sin x & 1 & x^2 + e^x \\ x - \cos x & x^2 - e^x & 4x^4 - 8x^3 + 7x^2 + 5 \end{pmatrix}$ .

Define $f(x) = \det\left(\frac{A(x) + A^T(x)}{2}\right) + \det\left(\frac{A(x) - A^T(x)}{2}\right)$ .

If $S$ is the sum of all local extreme values of the function $f(x)$ , then the value of $S$ is:

Options

A.
5
B.
11
C.
6
D.
16

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

Properties of Transpose Determinant|Symmetric and Skew-Symmetric Decomposition|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the properties of determinants for symmetric and skew-symmetric matrices, combined with a classic calculus trap.

Students must first recognize that the determinant of an odd-order skew-symmetric matrix is zero, simplifying the function to the determinant of the symmetric part.

The primary trap lies in the second derivative test: the critical point at x=1 yields f''(1)=0.

Students who blindly assume all critical points are extrema (or fail to check the sign change of the first derivative) will incorrectly include f(1)=6 in their sum, leading to the trap answer 11.

The true nature of x=1 is a point of inflection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Properties of Transpose Determinant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free