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Question #40415

Difficulty: 8/10 | 🟢 Free

The Question

A curve is defined implicitly by the equation $x = 3\sqrt{2}y + |\sin 3y - \sin y|$ for $y \in \left(0, \frac{\pi}{2}\right)$ . Let $y = g(x)$ be the continuous function defined by this curve. If $x_0 = \frac{3\sqrt{2}\pi}{4}$ , and the left-hand and right-hand derivatives of $g(x)$ at $x_0$ are denoted by $L = \lim_{x \to x_0^-} g'(x)$ and $R = \lim_{x \to x_0^+} g'(x)$ respectively, find the value of $\frac{1}{L \cdot R}$ .

Options

A.
8
B.
10
C.
12
D.
14

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

Factor-Product Conversion|Derivative of Implicit Inverse Forms|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the derivative of an implicit inverse function combined with a modulus breakdown.

Students often incorrectly assume global differentiability or fail to identify the sign change of the modulus argument.

By requiring the factor-product conversion ( $\sin 3y - \sin y = 2\sin y \cos 2y$ ), the hidden non-differentiable point at $y = \pi/4$ is revealed.

The trap lies in evaluating the left and right derivatives of the inverse function, which forces the student to correctly apply $g'(x) = 1/x'(y)$ while managing the piecewise nature of the original function across the critical point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Factor-Product Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free