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Question #41311

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the unique real-valued continuous function satisfying the implicit equation $y^3 + y^2 \sin(|x|) + y = x^2 + 2$ for all $x \in \mathbb{R}$ . If $L = \lim_{x \to 0^+} f'(x)$ and $R = \lim_{x \to 0^-} f'(x)$ , then the value of $16|L \cdot R| + 8|L - R|$ is

Options

A.
1
B.
3
C.
5
D.
7

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Differentiability vs Continuity|Implicit Differentiation Rate|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' within implicit differentiation.

Students frequently differentiate terms like $\sin(|x|)$ incorrectly as $\cos(|x|)$ , ignoring the chain rule for $|x|$ which introduces a $\text{sgn}(x)$ multiplier.

This error leads to the false conclusion that $L = R = -1/4$ .

By evaluating the right-hand and left-hand limits separately, the problem highlights how the modulus transition at $x=0$ causes a jump discontinuity in the derivative, rendering the continuous function non-differentiable at that point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Differentiability vs Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free