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Question #40388

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{x^4 - 8x^3 + 24x^2 - 32x + 16}{x^4 + 16}$ Let $S$ be the range of $f$ . Which of the following represents the set $S$ ?

Options

A.
$[0, 8]$
B.
$(1, 8]$
C.
$[1, 8]$
D.
$[0, 1) \cup (1, 8]$

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

Codomain vs Range Distinction|Infinite Limit Behavior|Second Derivative Test Limitations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to determine the global range of a rational function by analyzing infinite limits and local extrema, specifically targeting the limitations of the Second Derivative Test.

At the critical point $x=2$ , the second derivative $f''(2)$ evaluates to zero.

A common trap is for students to incorrectly assume that $f''(c)=0$ implies an inflection point rather than a local extremum.

If they fall for this, they miss the global minimum at $x=2$ and rely solely on the infinite limit $\lim_{x \to \pm\infty} f(x) = 1$ to establish the lower bound of the range, leading them to incorrectly choose $(1, 8]$ or $[1, 8]$ .

Quick Facts About This Question

Question Information

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