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Question #41322

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued continuous function defined on its maximal domain $D \subset (0, 2\pi)$ satisfying the implicit equation $y^{\sin x} = (\sin x)^y$ with $y > 0$ . Let $x_0 \in D$ be the point where $f(x)$ attains its global maximum. Find the value of $\lim_{x \to x_0} \frac{f(x) - 1}{(2x - \pi)^2}$

Options

A.
$-\frac{1}{8}$
B.
$-\frac{1}{4}$
C.
$0$
D.
$\frac{1}{8}$

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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

Implicit Differentiation Implicit Domain|Logarithmic Differentiation|Symmetry and Periodicity

Why This Question Is Tricky

Common Mistakes

This question tests the lethal trap of blindly applying logarithmic differentiation to implicit functions without analyzing the implicit domain and injectivity constraints.

Students will typically take the log and differentiate implicitly, leading to a massive, circular algebraic expression for $y'$ that is extremely tedious to evaluate using L'Hopital's rule.

The key is to recognize that the base condition $\sin x > 0$ restricts the domain, and analyzing the logarithmic derivative function $g(t) = \frac{\ln t}{t}$ reveals it is strictly increasing on $(0, e]$ .

Since $\sin x \le 1$ , $y$ must also be in $(0, e)$ , forcing the unique explicit solution $y = \sin x$ .

Failing to simplify the function before taking the limit results in an insurmountable algebraic roadblock.

Quick Facts About This Question

Question Information

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