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
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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 \( h \)).
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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