Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.**](https://content.bartleby.com/qna-images/question/8e71c6f0-fdac-4001-9095-2485c121cf1d/05ca70f5-50a6-4e3e-80fc-c0885f2ccc21/k3yohhe_processed.jpeg)
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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