bounds -2 ≤ x ≤ 2π for 1 ≤ i ≤ n (here n = 2) n f(x) = x² + 25 Σ sin² Xi i=1 i=1 n
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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Question
Minimize the following function:
![### Function Definition
The function \( f(x) \) is defined as follows:
\[
f(x) = \sum_{i=1}^{n} x_i^2 + 25 \sum_{i=1}^{n} \sin^2 x_i
\]
### Details
- **Summation**: The expression contains two summations:
- The first summation \( \sum_{i=1}^{n} x_i^2 \) calculates the sum of the squares of the elements \( x_i \).
- The second summation \( 25 \sum_{i=1}^{n} \sin^2 x_i \) computes the sum of the squared sine values of the elements \( x_i \), each multiplied by 25.
### Constraints
- **Bounds**: The values of \( x_i \) are bounded by \( -2\pi \leq x_i \leq 2\pi \) for \( 1 \leq i \leq n \).
- **Specific Case**: In this scenario, \( n \) is equal to 2.
This function can be useful for analyzing periodic behaviors and quadratic interactions within the given range for different applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14458c04-f4d8-42cf-9f3b-e637b4a41ced%2F6aa85a1e-3aa6-4d60-9af7-98fb0edf38b7%2Fh04fbe2_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Definition
The function \( f(x) \) is defined as follows:
\[
f(x) = \sum_{i=1}^{n} x_i^2 + 25 \sum_{i=1}^{n} \sin^2 x_i
\]
### Details
- **Summation**: The expression contains two summations:
- The first summation \( \sum_{i=1}^{n} x_i^2 \) calculates the sum of the squares of the elements \( x_i \).
- The second summation \( 25 \sum_{i=1}^{n} \sin^2 x_i \) computes the sum of the squared sine values of the elements \( x_i \), each multiplied by 25.
### Constraints
- **Bounds**: The values of \( x_i \) are bounded by \( -2\pi \leq x_i \leq 2\pi \) for \( 1 \leq i \leq n \).
- **Specific Case**: In this scenario, \( n \) is equal to 2.
This function can be useful for analyzing periodic behaviors and quadratic interactions within the given range for different applications.
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