Use Fibonacci search method to minimize 𝑓(𝑥) = 𝑥
2 − 3𝑥 − 20 on [1,4] and 𝑛 = 3.

The Fibonacci search method is an optimization technique that leverages Fibonacci numbers to minimize a unimodal function over a given interval. Here, we are tasked with minimizing the function \( f(x) = x^2 - 3x - 20 \) on the interval \([1, 4]\) using 3 iterations (\(n = 3\)).

### Step 1: Compute Fibonacci Numbers
First, we calculate the Fibonacci numbers \( F_0 \), \( F_1 \), \( F_2 \), and \( F_3 \):

\[
\begin{aligned}
F_0 & = 1 \\
F_1 & = 1 \\
F_2 & = 2 \\
F_3 & = 3
\end{aligned}
\]

### Step 2: Compute the Points
The formula for the points in the interval is given by:

\[
x_1 = a + \frac{F_{n-2}}{F_n}(b - a)
\]
\[
x_2 = a + \frac{F_{n-1}}{F_n}(b - a)
\]

For \(n = 3\) and the interval \([1, 4]\):

\[
x_1 = 1 + \frac{F_{1}}{F_{3}}(4 - 1) = 1 + \frac{1}{3}(3) = 1 + 1 = 2
\]
\[
x_2 = 1 + \frac{F_{2}}{F_{3}}(4 - 1) = 1 + \frac{2}{3}(3) = 1 + 2 = 3
\]

### Step 3: Evaluate the Function at \(x_1\) and \(x_2\)
Now, we evaluate the function at \(x_1\) and \(x_2\):

\[
f(x_1) = f(2) = 2^2 - 3(2) - 20 = 4 - 6 - 20 = -22
\]

\[
f(x_2) = f(3) = 3^2 - 3(3) - 20 = 9 - 9 - 20 = -20
\]

### Step 4: Determine the New Interval
Since \( f(x_1) < f(x_2) \), the new interval is \([1, 3]\).

### Step 5: Next Iteration
For \(n = 2\), compute the new points \(x_1'\) and \(x_2'\) in the interval \([1, 3]\):

\[
x_1' = 1 + \frac{F_{0}}{F_{2}}(3 - 1) = 1 + \frac{1}{2}(2) = 1 + 1 = 2
\]
\[
x_2' = 1 + \frac{F_{1}}{F_{2}}(3 - 1) = 1 + \frac{1}{2}(2) = 1 + 1 = 2
\]

Both \(x_1'\) and \(x_2'\) are at the same point \(x = 2\).

Since the interval has effectively collapsed to a single point at \(x = 2\), and \(f(x)\) is minimized there, we conclude:

### Conclusion
The function \( f(x) = x^2 - 3x - 20 \) is minimized at \( x = 2 \), with a minimum value of \( f(2) = -22 \).

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Would you like to explore the derivation in more detail or have any questions about the process?

### Related Questions:
1. What is the general formula for Fibonacci numbers and how is it derived?
2. How does the Fibonacci search method compare to the golden-section search?
3. What are the advantages of using the Fibonacci search method over other optimization techniques?
4. How can the Fibonacci search method be adapted for functions with multiple local minima?
5. How does the choice of \(n\) affect the accuracy of the Fibonacci search method?

**Tip:** The Fibonacci search method is particularly efficient for unimodal functions, where it systematically narrows the search interval based on the Fibonacci sequence.

Explore how to minimize the function f(x) = x^2 - 3x - 20 over the interval [1, 4] using the Fibonacci search method with 3 iterations. Follow a step-by-step solution and understand the application of Fibonacci numbers in optimization. Suitable for advanced undergraduate students and anyone interested in optimization techniques.

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