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Find a counterexample to show that the statement is false. (Select all that apply.) 1 X x>- Ox>2 ✔x>1 Ox≤-1 none of these D0

Find a counterexample to show that the statement is false. (Select all that apply.) 1 X x>- Ox>2 ✔x>1 Ox≤-1 none of these D0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Which options shows that the statement is false? Finding a counterexample
**Problem Statement:** Find a counterexample to show that the statement is false. (Select all that apply.) \( x > \frac{1}{x} \) **Options:** - [ ] \( x > 2 \) - [x] \( x > 1 \) - [ ] \( x \leq -1 \) - [ ] none of these - [ ] \( 0 < x < 1 \) - [x] \( -1 < x < 0 \) A red "X" indicates the selections are incorrect. **Additional Information:** Below the options, there is a prompt asking, "Need Help?" followed by a button labeled "Read It."
Transcribed Image Text:**Problem Statement:** Find a counterexample to show that the statement is false. (Select all that apply.) \( x > \frac{1}{x} \) **Options:** - [ ] \( x > 2 \) - [x] \( x > 1 \) - [ ] \( x \leq -1 \) - [ ] none of these - [ ] \( 0 < x < 1 \) - [x] \( -1 < x < 0 \) A red "X" indicates the selections are incorrect. **Additional Information:** Below the options, there is a prompt asking, "Need Help?" followed by a button labeled "Read It."
Expert Solution
Step 1

Given x>1x.

This can be written as:

x>1x⇒x2>1

Consider the equality x2=1.

Solving this:

x2=1⇒x=±1

Also 1x has a denominator equal to 0, at x=0.

So the critical points are x=-1, 0, 1.

So, there can be three intervals (-∞,-1) , (-1,0), 0,1 ,(1,∞).

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