12. Sketch the interval (a,b) on the x-axis with the point c inside. Then find the largest value of 8>0 such that for all x, 0A
12. Sketch the interval (a,b) on the x-axis with the point c inside. Then find the largest value of 8>0 such that for all x, 0A
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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Question
![**12.** Sketch the interval \( (a, b) \) on the x-axis with the point \( c \) inside. Then find the largest value of \(\delta > 0\) such that for all \( x \), \( 0 < |x - c| < \delta \) implies \( a < x < b \).
\[ a = \frac{1}{3}, \, b = \frac{5}{9}, \, c = \frac{1}{2} \]
Choose the correct sketch below.
**Option A:**
- Number line from 0 to 1.
- Interval starts at \(1/3\) and ends at \(5/9\), inclusive (\(<-|\text{ }|->\)).
- Point \(0\) at the start, and point \(1\) at the end.
**Option C:**
- Number line from 0 to 1.
- Interval starts at \(1/3\) and ends at \(5/9\), inclusive (\(<-|\text{ }|->\)), with a point at \(1/2\).
- Labels for 0, \(1/3\), \(1/2\), \(5/9\), and 1.
The largest possible value for \(\delta\) is \(\boxed{\frac{1}{6}}\). (Type a simplified fraction.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6f5a197-bd68-4fd1-8232-f49c284d6121%2F7ca52cd5-7dca-45ba-a9d9-10f6a075e33c%2Fhucppp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**12.** Sketch the interval \( (a, b) \) on the x-axis with the point \( c \) inside. Then find the largest value of \(\delta > 0\) such that for all \( x \), \( 0 < |x - c| < \delta \) implies \( a < x < b \).
\[ a = \frac{1}{3}, \, b = \frac{5}{9}, \, c = \frac{1}{2} \]
Choose the correct sketch below.
**Option A:**
- Number line from 0 to 1.
- Interval starts at \(1/3\) and ends at \(5/9\), inclusive (\(<-|\text{ }|->\)).
- Point \(0\) at the start, and point \(1\) at the end.
**Option C:**
- Number line from 0 to 1.
- Interval starts at \(1/3\) and ends at \(5/9\), inclusive (\(<-|\text{ }|->\)), with a point at \(1/2\).
- Labels for 0, \(1/3\), \(1/2\), \(5/9\), and 1.
The largest possible value for \(\delta\) is \(\boxed{\frac{1}{6}}\). (Type a simplified fraction.)
Expert Solution
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We have to sketch the interval and find the largest value of delta.
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