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Question #40332

Difficulty: 8/10 | 🟢 Free

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = (|x-1| - 3) e^{-|x|}$ . The range of $f(x)$ is:

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

Domain and Range Nuance|Increasing/Decreasing Strictness|First Derivative Test Trap

This question tests the 'First Derivative Test Trap' combined with domain nuances.

Students often rely solely on solving $f'(x) = 0$ to find extrema, which yields $x = -3$ and $x = 5$ , leading to an incorrect range.

By introducing absolute value functions, critical points also occur where the derivative is undefined (cusps at $x=0$ and $x=1$ ).

The global minimum actually occurs at the non-differentiable point $x=0$ .

Failing to partition the domain and check these singular points results in missing the true bounds of the function's range.

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Domain and Range Nuance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free