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Question #40443

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \sin x & \cos x & e^{\sin x} \\ \cos x & -\sin x & e^{\sin x}\cos x \\ -\sin x & -\cos x & e^{\sin x}(\cos^2 x - \sin x) \end{vmatrix}$ . Find the number of critical points of $f(x)$ in the interval $[0, 2\pi]$ .

Options

A.
4
B.
2
C.
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

Monotonicity and Boundedness|Differentiation of Determinants|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

The question tests the evaluation of a Wronskian-like determinant and the subsequent monotonicity analysis.

The primary trap is the 'Singular Threshold' fallacy: students often assume that the critical points of a determinant function $f(x) = \det(A(x))$ occur exclusively where the matrix becomes singular ( $f(x) = 0$ ).

Here, $\det(A) = 0$ simplifies to $\sin^2 x + \sin x - 2 = 0$ , which only yields $x = \pi/2$ .

Relying solely on this threshold causes students to miss the other three critical points generated by the chain rule and the embedded quadratic $\sin^2 x + 3\sin x - 1 = 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Monotonicity and Boundedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free