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Question #40950

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = 12|x^2 - 4| + 4|x|^3 - 30x^2 + 72|x|$ . If $S$ is the set of all points of local extrema of $f$ , then the value of $\sum_{x \in S} 2^{|x|}$ is equal to

Options

A.
13
B.
12
C.
5
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Algebra of Derivatives|Non-Differentiable Extrema|Symmetry Utilization

Why This Question Is Tricky

Common Mistakes

This question tests the identification of local extrema at non-differentiable points, a classic JEE Advanced trap.

Students often mechanically apply Fermat's Theorem ( $f'(x)=0$ ) and find only the smooth extrema ( $x=\pm 1$ ), completely missing the cusps/corners at $x=\pm 2$ and $x=0$ where the derivative is undefined but the Extreme Value Theorem still applies.

The use of an even function mandates symmetry utilization to halve the algebraic workload, while the specific sum $\sum 2^{|x|}$ uniquely penalizes the omission of the origin ( $x=0$ ), which is frequently overlooked as a boundary condition of the absolute value.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebra of Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free