Given fiRn DR, fecl, Xıkl ER", and let thele exist de fine a symmetric positive definite Matrix N ERAxa $ ;R-DR by $(x)=f(x(k! _XNXf(x"}) XER, and f(x(kl) +0. assumed y Proof that $1(0) <0, Justify your answer.
Given fiRn DR, fecl, Xıkl ER", and let thele exist de fine a symmetric positive definite Matrix N ERAxa $ ;R-DR by $(x)=f(x(k! _XNXf(x"}) XER, and f(x(kl) +0. assumed y Proof that $1(0) <0, Justify your answer.
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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![### Mathematical Problem Statement
Given \( f:\mathbb{R}^n \rightarrow \mathbb{R} \), \( f \in C^1 \), \( x^{(k)} \in \mathbb{R}^n \), and let there exist a symmetric positive definite matrix \( N \in \mathbb{R}^{n \times n} \).
Define \( \phi : \mathbb{R} \rightarrow \mathbb{R} \) by
\[
\phi(x) = f(x^{(k)} - \alpha N \nabla f(x^{(k)}))
\]
Assume \( \alpha \in \mathbb{R} \), and \( \nabla f(x^{(k)}) \neq 0 \).
1. **Proof that \( \phi(0) < 0 \). Justify your answer.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e71c6f0-fdac-4001-9095-2485c121cf1d%2F05ca70f5-50a6-4e3e-80fc-c0885f2ccc21%2Fk3yohhe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Problem Statement
Given \( f:\mathbb{R}^n \rightarrow \mathbb{R} \), \( f \in C^1 \), \( x^{(k)} \in \mathbb{R}^n \), and let there exist a symmetric positive definite matrix \( N \in \mathbb{R}^{n \times n} \).
Define \( \phi : \mathbb{R} \rightarrow \mathbb{R} \) by
\[
\phi(x) = f(x^{(k)} - \alpha N \nabla f(x^{(k)}))
\]
Assume \( \alpha \in \mathbb{R} \), and \( \nabla f(x^{(k)}) \neq 0 \).
1. **Proof that \( \phi(0) < 0 \). Justify your answer.**
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