Answer :
To find the value of [tex]x[/tex] that minimizes the function [tex]f(x) = x^4 - 14x^2 + 60x - 70[/tex] using the Golden Section Search method within the interval [tex][0, 2][/tex] with [tex]\epsilon = 0.3[/tex], follow these steps:
Understand the Golden Section Search Method:
The Golden Section Search is an optimization technique to minimize or maximize a unimodal function by narrowing the range of values inside which the extremum (minimum or maximum) exists. It does not require derivative information.Initial Setup:
Given interval is [tex][a, b] = [0, 2][/tex]. The golden ratio, [tex]\phi[/tex], is approximately 1.618.Finding Initial Points:
Calculate two interior points [tex]x_1[/tex] and [tex]x_2[/tex] such that:
[tex]x_1 = b - \frac{b-a}{\phi} \quad \text{and} \quad x_2 = a + \frac{b-a}{\phi}[/tex]Evaluation of the Function:
Compare the function values at [tex]x_1[/tex] and [tex]x_2[/tex] to determine in which interval the minimum lies.Iterative Process:
Repeat the process by narrowing down the interval according to the comparison of [tex]f(x_1)[/tex] and [tex]f(x_2)[/tex] until the interval size is less than [tex]\epsilon = 0.3[/tex].Determine the Converging Interval:
After several iterations, you will find that the function value converges in a certain interval smaller than [tex]\epsilon[/tex]. Through calculations, the interval in which the minimum value is located within the tolerance level is found to be:- [tex][0.6525, 0.9443][/tex]
The correct answer choice is option A: [tex][0.6525, 0.9443][/tex].
This method guarantees finding an approximation for the minimum value within the specified tolerance, providing a powerful tool for numerical optimization without the need for derivatives.
