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Question #41315

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = |x^2 - 4|^{|x-2|} + |x^2 - 2x + 2|^{|x^2 - 1|} + |x^2 - 9|^{|x+3|}$ Let $S$ be the set of all points $x \in \mathbb{R}$ where $f(x)$ is not differentiable. Find the sum of all elements in $S$ .

Options

A.
$-2$
B.
$-1$
C.
$0$
D.
$2$

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 Domain|Modulus Sign Dependency|Modulus Inequality Logic|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

The question constructs a 'Lethal Collision' by intertwining logarithmic differentiation domains with modulus piecewise dependencies.

Students typically memorize that $|f(x)|$ is non-differentiable at its roots, but for $|f(x)|^{g(x)}$ , the differentiability depends heavily on the limiting behavior of the exponent.

The trap exposes three distinct fallacies: assuming $|0|^k$ is always non-differentiable (it is differentiable if $k>1$ ), assuming $|0|^0$ is differentiable because it evaluates to 1, and failing to check the base value when the exponent is non-differentiable (if the base is 1, the left/right derivative mismatch is nullified by $\ln(1)=0$ ).

Quick Facts About This Question

Question Information

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