To solve the problem of finding the minimum of the function $f(x) = x(x - 1.5)$ in the interval $[0, 1]$ using the Fibonacci search algorithm with $n = 4$, we will follow these steps:

Fibonacci Numbers

We need the first five Fibonacci numbers (for $n = 4$):

$F_0 = 0, \, F_1 = 1, \, F_2 = 1, \, F_3 = 2, \, F_4 = 3, \, F_5 = 5$

The length of the search interval will be divided according to the Fibonacci ratios.

Step-by-step process:

1. Initial Interval: $[a_0, b_0] = [0, 1]$

   Calculate two test points based on the Fibonacci sequence: $x_1 = a_0 + \frac{F_3}{F_5} \cdot (b_0 - a_0) = 0 + \frac{2}{5} \cdot (1 - 0) = 0.4$ $x_2 = a_0 + \frac{F_4}{F_5} \cdot (b_0 - a_0) = 0 + \frac{3}{5} \cdot (1 - 0) = 0.6$

2. Evaluate Function: $f(x_1) = f(0.4) = 0.4(0.4 - 1.5) = 0.4 \times (-1.1) = -0.44$ $f(x_2) = f(0.6) = 0.6(0.6 - 1.5) = 0.6 \times (-0.9) = -0.54$ Since $f(x_2) < f(x_1)$, the minimum lies between $x_1$ and $b_0$, so we update the interval to: [ [a_1, b_1] = [0.4, 1] ]

3. Next Points: Using Fibonacci ratios, we compute new test points. Since $x_2$ is already inside the interval, we calculate only the new point: $x_3 = a_1 + \frac{F_2}{F_4} \cdot (b_1 - a_1) = 0.4 + \frac{1}{3} \cdot (1 - 0.4) = 0.4 + 0.2 = 0.6$ (This value coincides with $x_2$, so $x_2$ remains unchanged.)

4. Evaluate Function Again: We already know: $f(x_2) = -0.54$ Now calculate for $x_3 = 0.4$ (already done): $f(x_3) = -0.44$ Since $f(x_2) < f(x_3)$, the interval shrinks to $[x_1, b_1]$.

Thus, we conclude that the minimum of $f(x) = x(x - 1.5)$ is at $x_2 = 0.6$ after 4 iterations of Fibonacci search.

Final Result:

The minimum of $f(x)$ on the interval $[0, 1]$ using Fibonacci search is approximately at $x = 0.6$.

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Do you want more details or have any questions about the Fibonacci search?

Here are 5 related questions:

1. How is the Fibonacci sequence applied in optimization problems?
2. What is the difference between Fibonacci search and golden-section search?
3. Can the Fibonacci method be applied to multi-dimensional optimization problems?
4. How does the accuracy of Fibonacci search improve with larger $n$?
5. What are the limitations of the Fibonacci search method in practical applications?

Tip: Fibonacci search is especially useful when function evaluations are costly, as it minimizes the number of function calls.