2. Let f [a, b] →→R be continuously differentiable on [a, b] with f(a) = f(b) = 0 and : Prove that - f² = 1. rb [*27(e)f(x) dx = -_and_ (/[^\P(x) dx}) ({{* ²ª\ƒ©@Fªdz) > } -
2. Let f [a, b] →→R be continuously differentiable on [a, b] with f(a) = f(b) = 0 and : Prove that - f² = 1. rb [*27(e)f(x) dx = -_and_ (/[^\P(x) dx}) ({{* ²ª\ƒ©@Fªdz) > } -
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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![**Problem Statement:**
Let \( f : [a, b] \to \mathbb{R} \) be continuously differentiable on \([a, b]\) with \( f(a) = f(b) = 0 \) and
\[
\int_a^b f^2 = 1.
\]
Prove that
\[
\int_a^b x f(x) f'(x) \, dx = -\frac{1}{2}
\]
and
\[
\left( \int_a^b [f'(x)]^2 \, dx \right) \left( \int_a^b x^2 [f(x)]^2 \, dx \right) > \frac{1}{4}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabba7d05-e030-4d49-ac3b-b588659cd1ab%2Fdf8e0257-3b6c-4ff8-86bc-5b3ffdac2049%2Fcf10tzp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( f : [a, b] \to \mathbb{R} \) be continuously differentiable on \([a, b]\) with \( f(a) = f(b) = 0 \) and
\[
\int_a^b f^2 = 1.
\]
Prove that
\[
\int_a^b x f(x) f'(x) \, dx = -\frac{1}{2}
\]
and
\[
\left( \int_a^b [f'(x)]^2 \, dx \right) \left( \int_a^b x^2 [f(x)]^2 \, dx \right) > \frac{1}{4}.
\]
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