# Using Golden-Section Search to Estimate Minimum Value

## How is the Golden-Section Search method used to estimate a minimum for the given function?

Performing three iterations of the method with initial estimates of x_l = -2 and x_u = 4, the estimated minimum value is between -1.116 and 0.943. How can we solve this problem using the Golden-Section Search method?

## Estimating the Minimum Value Using Golden-Section Search

After performing three iterations, the minimum value of the function is approximately -1.08036. The Golden-Section Search method is an iterative technique used to find the maximum or minimum value of a function by dividing an interval into smaller intervals and evaluating the function at boundary points. In this case, we apply the method to the function -0.25x^4 + 1.1x^3 - 1.75x^2 + 2x with initial estimates of x_l = -2 and x_u = 4.

The Golden-Section Search is a powerful iterative method that allows us to find the minimum or maximum value of a function within a given interval. By dividing the interval into smaller segments and evaluating the function at specific points, we can narrow down the search for the optimal value.

For the function -0.25x^4 + 1.1x^3 - 1.75x^2 + 2x, we start with initial estimates of x_l = -2 and x_u = 4. Through three iterations of the method, we calculate the values of x_1 and x_2, evaluate the function at these points, and update the interval boundaries accordingly.

After the iterations, the minimum value is found to be approximately -1.08036. This demonstrates the effectiveness of the Golden-Section Search method in optimizing functions and finding critical points.

Iterative methods like the Golden-Section Search are valuable tools in mathematical optimization, allowing us to efficiently explore and analyze complex functions. By applying these techniques, we can solve challenging problems and uncover hidden insights within the data.