Use the golden section search to find the value of x that minimizes f(x)=x4-14x3+60x2-70x in the range [0,2]. locate this value of x to within a range of 0.3



To apply the Golden Section Search to minimize the function \( f(x) = x^4 - 14x^3 + 60x^2 - 70x \) over the range [0, 2], we proceed with the following steps.

### Steps in the Golden Section Search:

1. **Initialization**:
   - Define the range as \([a, b] = [0, 2]\).
   - The golden ratio constant is \( \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618 \).

2. **Choose two interior points** within the interval \([a, b]\) based on the golden ratio:
   - \(x_1 = b - \frac{b - a}{\phi}\)
   - \(x_2 = a + \frac{b - a}{\phi}\)
   
   Evaluate the function \( f(x_1) \) and \( f(x_2) \).

3. **Iterative narrowing**:
   - If \( f(x_1) < f(x_2) \), then the minimum lies in the interval \([a, x_2]\), so update \( b = x_2 \).
   - If \( f(x_1) > f(x_2) \), then the minimum lies in the interval \([x_1, b]\), so update \( a = x_1 \).
   - Repeat this process until the interval width \([b - a]\) is within the desired precision (0.3 in this case).

4. **Stopping condition**:
   - The process continues until the interval \([a, b]\) is less than 0.3, at which point \( \frac{a + b}{2} \) will be taken as the approximation for the minimum value of \(x\).

Let's carry out the calculations step-by-step to locate the value of \( x \) that minimizes \( f(x) \) within the range of [0, 2] and tolerance 0.3.The value of \( x \) that approximately minimizes \( f(x) = x^4 - 14x^3 + 60x^2 - 70x \) within the range [0, 2] is \( x \approx 0.798 \), located to within the desired tolerance of 0.3.

Do you have any questions, or would you like further details?

Here are 5 related questions to expand on this:
1. What is the significance of the golden ratio in optimization techniques?
2. How does the Golden Section Search compare to other optimization methods like Newton's method?
3. Could you apply the same method to maximize a function instead of minimizing it?
4. How does changing the tolerance value affect the precision and computation time?
5. Can the Golden Section Search be applied to functions with multiple local minima?

**Tip**: The Golden Section Search is especially useful for unimodal functions, where the function has only one minimum in the interval of interest.

Use the golden section search to find the value of x that minimizes f(x)=x^4-14x^3+60x^2-70x in the range [0,2]. Locate this value of x to within a range of 0.3.

Learn how to use the Golden Section Search to minimize the function f(x)=x^4 - 14x^3 + 60x^2 - 70x within the range [0, 2], and locate the value of x with a tolerance of 0.3. This optimization problem explores key mathematical concepts such as the golden ratio and polynomial functions.

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