Find the minimum of 𝑓(𝑥) = 𝑥(𝑥 − 1.5) in [0, 1] by Golden section search rule with
interval of uncertainty as 0.3

The Golden Section Search is an iterative method used to find the minimum (or maximum) of a unimodal function over a specific interval. In your case, the function is:

\[
f(x) = x(x - 1.5)
\]

We aim to find the minimum of \( f(x) \) in the interval \([0, 1]\) with an interval of uncertainty of \(0.3\), meaning we are looking for the point where the uncertainty of the interval is reduced to less than or equal to 0.3.

The steps for the Golden Section Search method are as follows:

### Step 1: Set the interval \([a, b]\)

Initial interval: \([a, b] = [0, 1]\).

### Step 2: Define the golden ratio constant

The golden ratio constant \( \phi \) is approximately \( 1.618 \), and the inverse of the golden ratio is:

\[
r = \frac{1}{\phi} \approx 0.618
\]

### Step 3: Calculate the interior points

Let \( c \) and \( d \) be the interior points of the interval \([a, b]\). These are calculated as:

\[
c = a + (1 - r)(b - a)
\]
\[
d = a + r(b - a)
\]

Substitute the values \( a = 0 \) and \( b = 1 \):

\[
c = 0 + (1 - 0.618)(1 - 0) = 0.382
\]
\[
d = 0 + 0.618(1 - 0) = 0.618
\]

### Step 4: Evaluate the function at the points \( c \) and \( d \)

We now evaluate \( f(x) \) at the points \( c = 0.382 \) and \( d = 0.618 \).

\[
f(c) = 0.382(0.382 - 1.5) = 0.382(-1.118) = -0.427
\]
\[
f(d) = 0.618(0.618 - 1.5) = 0.618(-0.882) = -0.545
\]

### Step 5: Determine the next interval

Since \( f(d) < f(c) \), the minimum lies in the interval \([c, b] = [0.382, 1]\). We now repeat the process with this reduced interval.

### Step 6: Repeat until the interval of uncertainty is less than or equal to 0.3

For the next iteration, I'll perform the steps to reduce the interval further and ensure the uncertainty is within 0.3.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can continue following the Golden Section Search process manually by repeating the calculations until the interval is reduced to less than 0.3. Would you like me to explain the next steps for this or guide you through the final stage?

In the meantime, here are some related questions to help you explore further:

1. How can you implement Golden Section Search in programming languages like Python?
2. What are other methods to find minima of functions within an interval, like the bisection method or Newton's method?
3. Can the Golden Section Search be applied to functions with multiple local minima?
4. How does the uncertainty parameter impact the accuracy and speed of the Golden Section Search?
5. What are the limitations of Golden Section Search compared to other optimization techniques?

**Tip**: Always verify the interval reduction carefully to ensure your target uncertainty is achieved.

Find the minimum of 𝑓(𝑥) = 𝑥(𝑥 − 1.5) in [0, 1] by Golden section search rule with interval of uncertainty as 0.3

Explore how to apply the Golden Section Search method to find the minimum of f(x) = x(x - 1.5) within the interval [0, 1]. This step-by-step guide includes key formulas, the use of the golden ratio, and reducing the uncertainty to 0.3.

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