Let the function h: R → R be bounded. Define the function f: R→ R by f(x) = 1+ 4x + x²h(x) for all x. Prove that f(0) = 1 and f'(0) = 4. (Note: There is no assumption about the differentiability of the function h.
Let the function h: R → R be bounded. Define the function f: R→ R by f(x) = 1+ 4x + x²h(x) for all x. Prove that f(0) = 1 and f'(0) = 4. (Note: There is no assumption about the differentiability of the function h.
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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![Let the function \( h: \mathbb{R} \to \mathbb{R} \) be bounded. Define the function \( f: \mathbb{R} \to \mathbb{R} \) by
\[
f(x) = 1 + 4x + x^2 h(x)
\]
for all \( x \).
Prove that \( f(0) = 1 \) and \( f'(0) = 4 \). (Note: There is no assumption about the differentiability of the function \( h \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80fac62b-1f91-4b94-88b2-6d2c841a4645%2F367cca7a-3214-48e5-98a0-0301fd09c26c%2Fdbhq4rn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let the function \( h: \mathbb{R} \to \mathbb{R} \) be bounded. Define the function \( f: \mathbb{R} \to \mathbb{R} \) by
\[
f(x) = 1 + 4x + x^2 h(x)
\]
for all \( x \).
Prove that \( f(0) = 1 \) and \( f'(0) = 4 \). (Note: There is no assumption about the differentiability of the function \( h \)).
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