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Question #40956

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be the real-valued continuous function defined implicitly by the equation $y^5 + y^3 + y = |x^2 - 4|^{|x-1|}$ for all $x \in \mathbb{R}$ . Let $S$ be the set of all points $x \in \mathbb{R}$ where $y(x)$ is not differentiable. Find the sum of all elements in $S$ .

Options

A.
1
B.
3
C.
-1
D.
0

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

Logarithmic Differentiation|Chain Rule Inner Function Differentiability|Implicit Differentiation Implicit Domain|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question targets the 'Lethal Collision' trap in logarithmic differentiation of complex modulus functions.

Students commonly apply the formula $\frac{d}{dx}\ln|f(x)| = \frac{f'(x)}{f(x)}$ blindly, which produces apparent singularities at the roots of $f(x)$ (here, $x = \pm 2$ ) and the roots of the exponent (here, $x = 1$ ).

The trap assumes all such critical points are non-differentiable.

However, the chain rule's inner function differentiability must be evaluated via first principles at roots.

At $x=-2$ , the higher exponent power ( $|-2-1|=3$ ) suppresses the singularity, making the function differentiable with a derivative of 0.

At $x=2$ , the exponent is 1, leading to a sharp corner (derivatives $\pm 4$ ).

Failing to manually check these limits causes students to incorrectly include $x=-2$ in the non-differentiable set, leading to the trap answer of 1.

Quick Facts About This Question

Question Information

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