Advanced Engineering Mathematics
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.**
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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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