14. Discuss the differentiability at x = 0 of the function 1 x sin -, x = 0, x f: x→ 0, x = 0.
14. Discuss the differentiability at x = 0 of the function 1 x sin -, x = 0, x f: x→ 0, x = 0.
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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![### Differentiability at \( x = 0 \)
#### Problem 14
Discuss the differentiability at \( x = 0 \) of the function:
\[
f : x \rightarrow
\begin{cases}
x \sin \frac{1}{x}, & x \neq 0, \\
0, & x = 0.
\end{cases}
\]
#### Problem 15
Discuss the differentiability at \( x = 0 \) of the function:
\[
f : x \rightarrow
\begin{cases}
x^n \sin \frac{1}{x}, & x \neq 0, \\
0, & x = 0.
\end{cases}
\]
where \( n \) is an integer larger than 1. For what values of \( k \), does the \( k \)th derivative exist at \( x = 0 \)? (See Problem 12.)
### Section 4.1: The Derivative in \( \mathbb{R}^1 \)
(Page 93)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F1cf1b3a4-6cd7-4d23-936c-3a006b127d05%2Fjbzztv7_processed.png&w=3840&q=75)
Transcribed Image Text:### Differentiability at \( x = 0 \)
#### Problem 14
Discuss the differentiability at \( x = 0 \) of the function:
\[
f : x \rightarrow
\begin{cases}
x \sin \frac{1}{x}, & x \neq 0, \\
0, & x = 0.
\end{cases}
\]
#### Problem 15
Discuss the differentiability at \( x = 0 \) of the function:
\[
f : x \rightarrow
\begin{cases}
x^n \sin \frac{1}{x}, & x \neq 0, \\
0, & x = 0.
\end{cases}
\]
where \( n \) is an integer larger than 1. For what values of \( k \), does the \( k \)th derivative exist at \( x = 0 \)? (See Problem 12.)
### Section 4.1: The Derivative in \( \mathbb{R}^1 \)
(Page 93)
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