### Mathematical Problem StatementGiven \( f:\mathbb{R}^n \rightarrow \mathbb{R} \), \( f \in C^1 \), \( x^{(k)} \in \mathbb{R}^n \), 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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Answered: Given fiRn DR, fecl, Xıkl ER", and let…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 25EQ

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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.