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Question #41681

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{vmatrix} \ln(1+x^2) & |x^2-3| & 1 \\ |x^2-3| & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}$ Let $S$ be the set of all points of local extrema (both local maxima and local minima) of the function $f(x)$ . Find the value of $\sum_{x \in S} x^2$ .

Options

A.
14
B.
16
C.
18
D.
20

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

Objective Function Gradient|Differentiation of Determinants|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to strategically evaluate the gradient of an objective function defined by a determinant.

The primary trap is attempting to differentiate the determinant directly using Jacobi's formula, which leads to a convoluted matrix derivative and mishandling of the chain rule on the modulus function $|x^2-3|$ at its roots $x=\pm\sqrt{3}$ .

By expanding the determinant first, the seemingly complex $\ln(1+x^2)$ entry cancels out, reducing the problem to analyzing the critical points of a piecewise function.

The 'Lethal Collision' occurs because students often ignore points of non-differentiability when finding extrema; here, the roots of the modulus function actually form sharp local minima that must be identified via first principles or the first derivative test.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Objective Function Gradient
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free