S) Employ the following methods to find the maximum of f(x)=4x-1.8x² +1.2x -0.3x (a) Golden-section search (x; = -2, x, = 4, ɛ; = 1%). (b)Parabolic interpolation (xo = 1.75, n = 2, x2 = 2.5, iterations = 4). Select new points sequentially as in the secant method. (c) Newton's method (x, = 3, E; = 1%).
S) Employ the following methods to find the maximum of f(x)=4x-1.8x² +1.2x -0.3x (a) Golden-section search (x; = -2, x, = 4, ɛ; = 1%). (b)Parabolic interpolation (xo = 1.75, n = 2, x2 = 2.5, iterations = 4). Select new points sequentially as in the secant method. (c) Newton's method (x, = 3, E; = 1%).
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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![**Problem 8:**
Utilize the following methods to determine the maximum of the function:
\[ f(x) = 4x - 1.8x^2 + 1.2x^3 - 0.3x^4 \]
**(a) Golden-section search**
- Initial lower bound (\(x_l\)): -2
- Initial upper bound (\(x_u\)): 4
- Stopping criterion (\(\varepsilon_s\)): 1%
**(b) Parabolic interpolation**
- Initial points:
- \(x_0 = 1.75\)
- \(x_1 = 2\)
- \(x_2 = 2.5\)
- Number of iterations: 4
- New points are selected sequentially, similar to the secant method.
**(c) Newton’s method**
- Initial guess (\(x_0\)): 3
- Stopping criterion (\(\varepsilon_s\)): 1%
This problem involves applying numerical optimization techniques to find the maximum value of a given polynomial function. Each method provides a different approach to approximating the maximum within a specified tolerance level.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4e23d50-b6e1-4880-9ad9-a789799b751b%2F980f004d-0c14-40aa-bdf4-bc63578649c0%2Fhxga0ta_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 8:**
Utilize the following methods to determine the maximum of the function:
\[ f(x) = 4x - 1.8x^2 + 1.2x^3 - 0.3x^4 \]
**(a) Golden-section search**
- Initial lower bound (\(x_l\)): -2
- Initial upper bound (\(x_u\)): 4
- Stopping criterion (\(\varepsilon_s\)): 1%
**(b) Parabolic interpolation**
- Initial points:
- \(x_0 = 1.75\)
- \(x_1 = 2\)
- \(x_2 = 2.5\)
- Number of iterations: 4
- New points are selected sequentially, similar to the secant method.
**(c) Newton’s method**
- Initial guess (\(x_0\)): 3
- Stopping criterion (\(\varepsilon_s\)): 1%
This problem involves applying numerical optimization techniques to find the maximum value of a given polynomial function. Each method provides a different approach to approximating the maximum within a specified tolerance level.
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