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Question #41178

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be a real-valued function defined implicitly by the equation $\sin^3(f(x)) + 2\sin(f(x)) = x^4 - 8x^2 + 12$ where $f(x) \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ . Let $S$ be the set of all $x \in \mathbb{R}$ at which $f(x)$ attains either its global maximum or its global minimum. Find the value of $\sum_{x \in S} x^2$ .

Options

A.
8
B.
16
C.
32
D.
64

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

Domain Constraint Intersections|Implicit to Explicit Conversion|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle implicit-to-explicit function conversion alongside strict domain constraints.

The core trap lies in the 'Second Derivative Test' and standard critical point analysis.

Students typically differentiate the polynomial $h(x) = x^4 - 8x^2 + 12$ to find extrema at $x=0, \pm 2$ .

However, the inverse trigonometric domain strictly bounds $h(x) \in [-3, 3]$ , completely excluding these critical points from the function's domain.

Consequently, the true global extrema occur exclusively at the non-differentiable boundary points of the domain, forcing students to abandon blind differentiation in favor of first-principle boundary analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Domain Constraint Intersections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free