To minimize the function $f(x) = x^3 - 6x^2 + 11x - 6$ using the Fibonacci search method, we need to follow these steps:

Step 1: Determine the Fibonacci numbers

First, we compute the Fibonacci numbers needed to achieve the required precision. The interval length must be reduced until it is less than 0.3, i.e., $\Delta x = 0.3$. The Fibonacci search algorithm reduces the uncertainty range in each iteration, and the total number of steps depends on the ratio between the initial interval size and the desired precision.

Given the initial interval $[1, 4]$, the total length is:

$L_0 = 4 - 1 = 3$

We need to find the smallest Fibonacci number $F_n$ such that:

$\frac{F_n}{F_{n+2}} \times 3 < 0.3$

By trial, we determine the Fibonacci numbers as follows:

$F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8, F_7 = 13, F_8 = 21, F_9 = 34, \ldots$

We compute the ratios to find when the interval length reduces below 0.3.

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Let me calculate the exact Fibonacci ratios for the interval.The relevant Fibonacci numbers are $F_3 = 2$, $F_4 = 3$, $F_5 = 5$, $F_6 = 8$, and $F_7 = 13$. The ratio $\frac{L_0}{F_7} = \frac{3}{13} \approx 0.231$, which satisfies the condition of reducing the uncertainty to less than 0.3.

Step 2: Perform the Fibonacci search

We will now apply the Fibonacci search method, iterating through steps and reducing the interval. At each iteration, we compute two points within the interval based on Fibonacci ratios, evaluate the function at those points, and then reduce the search interval accordingly.

We start with the interval $[1, 4]$, and proceed until the interval width is reduced below 0.3.

Let me calculate and display the iterations of the Fibonacci search for this problem.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can guide you through the Fibonacci search steps manually.

Would you like me to explain how to manually calculate the function evaluations and intervals at each step? Here are five questions that could deepen your understanding:

1. How does the Fibonacci search method differ from other optimization methods like the golden section search?
2. Why do we use Fibonacci ratios in this method?
3. How does the tolerance level affect the number of iterations in the search?
4. Can we apply the Fibonacci search method to functions that are not smooth or continuous?
5. What would be the stopping criterion if we required more precise minimization?

Tip: Fibonacci search is particularly efficient when the function evaluations are expensive because it minimizes the number of function calls.