Math Problem Statement

Suppose −2 ≤ f(x) ≤ x² − 6x + 2 for all x ≥ 2. Which of the following MUST be true? Select all correct answer choices.

Solution

The given problem asks to evaluate which of the following statements about a function f(x)f(x) are true, under the condition:

2f(x)x26x+2for allx2.-2 \leq f(x) \leq x^2 - 6x + 2 \quad \text{for all} \quad x \geq 2.

The choices are:

  1. limxf(x)=2\lim_{x \to \infty} f(x) = -2
  2. f(2)=2f(2) = 2
  3. limxf(x)=limxf(x)x\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{f(x)}{x}
  4. limxf(x)\lim_{x \to \infty} f(x) does not exist.

Analysis:

  1. Bounding f(x)f(x):

    • The function is bounded between 2-2 and x26x+2x^2 - 6x + 2, so as xx \to \infty, we need to analyze the bounds' behavior: limxx26x+2=,\lim_{x \to \infty} x^2 - 6x + 2 = \infty, which means f(x)f(x) \to \infty or could stay bounded within the range, depending on the nature of f(x)f(x).
  2. **First option

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Math Problem Analysis

Mathematical Concepts

Limits
Inequalities
Functions
Infinity Behavior

Formulas

Limit as x approaches infinity
Quadratic Function x² - 6x + 2

Theorems

Squeeze Theorem
Limit Comparison

Suitable Grade Level

Grades 11-12