24. If the second derivative f" exists at a value xo, show that f(xo + h) - 2f(xo) + f(xo − h) h² lim h→0 ƒ"(xo).
24. If the second derivative f" exists at a value xo, show that f(xo + h) - 2f(xo) + f(xo − h) h² lim h→0 ƒ"(xo).
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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24 pls
![**24.** If the second derivative \( f'' \) exists at a value \( x_0 \), show that
\[
\lim_{{h \to 0}} \frac{{f(x_0 + h) - 2f(x_0) + f(x_0 - h)}}{{h^2}} = f''(x_0).
\]
**25.** Suppose that \( f \) is defined by
\[
f: x \rightarrow
\begin{cases}
e^{-1/x^2} & \text{if } x > 0, \\
0 & \text{if } x \leq 0.
\end{cases}
\]
(a) Show that \( f^{(n)}(0) \) exists for every positive integer \( n \) (see Problem 7) and has the value 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cfc5ac-076f-4cb7-b69f-6b7f1cfee42f%2F66555bd8-ff44-43f5-9ca0-676661823d5d%2Fuqyv1j_processed.png&w=3840&q=75)
Transcribed Image Text:**24.** If the second derivative \( f'' \) exists at a value \( x_0 \), show that
\[
\lim_{{h \to 0}} \frac{{f(x_0 + h) - 2f(x_0) + f(x_0 - h)}}{{h^2}} = f''(x_0).
\]
**25.** Suppose that \( f \) is defined by
\[
f: x \rightarrow
\begin{cases}
e^{-1/x^2} & \text{if } x > 0, \\
0 & \text{if } x \leq 0.
\end{cases}
\]
(a) Show that \( f^{(n)}(0) \) exists for every positive integer \( n \) (see Problem 7) and has the value 0.
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