For this Optimization Problem: There's a function denoted as f: R -> R, and f(x) = (x-3)^4+ (x-3)^5. I want you to apply Newton's Method for finding a local minimizer that is applied to f. The initial point is given as x^(0) = 4. Let {x^(i)} C R for i ≥ 0 denote the sequence of points generated by Newton's Method. We assumed that this sequence converges to x*= 3. Please use rigorous proof to find the sequence {x^(i)} 's order of convergence.
For this Optimization Problem: There's a function denoted as f: R -> R, and f(x) = (x-3)^4+ (x-3)^5. I want you to apply Newton's Method for finding a local minimizer that is applied to f. The initial point is given as x^(0) = 4. Let {x^(i)} C R for i ≥ 0 denote the sequence of points generated by Newton's Method. We assumed that this sequence converges to x*= 3. Please use rigorous proof to find the sequence {x^(i)} 's order of convergence.
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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![### Optimization Problem: Newton's Method Application
For this Optimization Problem:
We have a function denoted as \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) = (x-3)^4 + (x-3)^5 \).
Your task is to apply Newton's Method for finding a local minimizer of this function \( f \). The initial point is given as \( x^{(0)} = 4 \).
Let \( \{x^{(i)}\} \subset \mathbb{R} \) for \( i \geq 0 \) denote the sequence of points generated by Newton's Method. It is assumed that this sequence converges to \( x^* = 3 \).
Please provide a rigorous proof to determine the order of convergence of the sequence \( \{x^{(i)}\} \).
### Explanation of the Given Function
- The function \( f(x) \) combines two terms: \( (x-3)^4 \) and \( (x-3)^5 \), which are both raised to the power of 4 and 5, respectively.
### Task
1. Initialize the sequence with \( x^{(0)} = 4 \).
2. Apply Newton's Method iteratively to generate the sequence \( \{x^{(i)}\} \).
3. Prove rigorously that the sequence converges to \( x^* = 3 \), and determine the order of convergence.
### Newton's Method
The general formula for Newton's Method for finding roots or extremum is:
\[ x^{(i+1)} = x^{(i)} - \frac{f'(x^{(i)})}{f''(x^{(i)})} \]
- \( f'(x) \) is the first derivative of \( f(x) \).
- \( f''(x) \) is the second derivative of \( f(x) \).
### Procedure
1. **Compute the first and second derivatives of \( f(x) \)**:
\[
f'(x) = \frac{d}{dx}[(x-3)^4 + (x-3)^5]
\]
\[
f''(x) = \frac{d^2}{dx^2}[(x-3)^4 + (x-3)^5]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e71c6f0-fdac-4001-9095-2485c121cf1d%2Fc2e3e99a-b065-48a1-af9c-5d140306b76f%2Fsmbrmml_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Optimization Problem: Newton's Method Application
For this Optimization Problem:
We have a function denoted as \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) = (x-3)^4 + (x-3)^5 \).
Your task is to apply Newton's Method for finding a local minimizer of this function \( f \). The initial point is given as \( x^{(0)} = 4 \).
Let \( \{x^{(i)}\} \subset \mathbb{R} \) for \( i \geq 0 \) denote the sequence of points generated by Newton's Method. It is assumed that this sequence converges to \( x^* = 3 \).
Please provide a rigorous proof to determine the order of convergence of the sequence \( \{x^{(i)}\} \).
### Explanation of the Given Function
- The function \( f(x) \) combines two terms: \( (x-3)^4 \) and \( (x-3)^5 \), which are both raised to the power of 4 and 5, respectively.
### Task
1. Initialize the sequence with \( x^{(0)} = 4 \).
2. Apply Newton's Method iteratively to generate the sequence \( \{x^{(i)}\} \).
3. Prove rigorously that the sequence converges to \( x^* = 3 \), and determine the order of convergence.
### Newton's Method
The general formula for Newton's Method for finding roots or extremum is:
\[ x^{(i+1)} = x^{(i)} - \frac{f'(x^{(i)})}{f''(x^{(i)})} \]
- \( f'(x) \) is the first derivative of \( f(x) \).
- \( f''(x) \) is the second derivative of \( f(x) \).
### Procedure
1. **Compute the first and second derivatives of \( f(x) \)**:
\[
f'(x) = \frac{d}{dx}[(x-3)^4 + (x-3)^5]
\]
\[
f''(x) = \frac{d^2}{dx^2}[(x-3)^4 + (x-3)^5]
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