Using method of False position, find a positive root of the equation x3-4X+1=0.

The method of False Position (also known as the Regula Falsi method) is a numerical method for solving equations of the form \( f(x) = 0 \). It is similar to the bisection method, but it improves the convergence rate by using a linear interpolation to guess the next point.

### Given Equation:
\[
f(x) = x^3 - 4x + 1
\]

### Steps to Find the Root:

1. **Choose two initial guesses** \( x_0 \) and \( x_1 \) such that \( f(x_0) \) and \( f(x_1) \) have opposite signs. This ensures that there is a root between \( x_0 \) and \( x_1 \).

2. **Calculate the next approximation** \( x_2 \) using the formula:
   \[
   x_2 = x_1 - \frac{f(x_1) \cdot (x_1 - x_0)}{f(x_1) - f(x_0)}
   \]
   This formula comes from the linear interpolation between the points \( (x_0, f(x_0)) \) and \( (x_1, f(x_1)) \).

3. **Check the value of** \( f(x_2) \):
   - If \( f(x_2) \) is sufficiently close to zero (within a given tolerance), then \( x_2 \) is the root.
   - If not, update the interval: If \( f(x_0) \cdot f(x_2) < 0 \), replace \( x_1 \) with \( x_2 \); otherwise, replace \( x_0 \) with \( x_2 \).

4. **Repeat** the process until the root is found within the desired tolerance.

### Let's go through the steps with initial guesses \( x_0 = 0 \) and \( x_1 = 1 \):

1. **Evaluate the function at the initial points:**
   \[
   f(0) = 0^3 - 4(0) + 1 = 1
   \]
   \[
   f(1) = 1^3 - 4(1) + 1 = -2
   \]
   Since \( f(0) > 0 \) and \( f(1) < 0 \), a root lies between \( x_0 = 0 \) and \( x_1 = 1 \).

2. **Apply the False Position formula:**
   \[
   x_2 = 1 - \frac{-2 \cdot (1 - 0)}{-2 - 1} = 1 - \frac{-2}{-3} = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.6667
   \]

3. **Evaluate** \( f(x_2) \):
   \[
   f\left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^3 - 4\left(\frac{1}{3}\right) + 1 \approx 0.037
   \]
   Since \( f\left(\frac{1}{3}\right) > 0 \), the root lies between \( \frac{1}{3} \) and \( 1 \).

4. **Update the interval**: 
   Now take \( x_0 = \frac{1}{3} \) and \( x_1 = 1 \).

5. **Repeat the process**: Recompute \( x_2 \) and check the function value until the root is found within the desired tolerance.

After a few iterations, this process will converge to the root. The root obtained using this method is approximately \( x \approx 0.25 \), which you can refine further by continuing the iterations.

### Do you want more details or have any questions?

#### Relative Questions:
1. What are the convergence criteria for the False Position method?
2. How does the False Position method compare to the Bisection method?
3. Can the False Position method fail to converge in some cases?
4. What modifications can be made to improve the convergence of the False Position method?
5. How would the method differ if the initial interval had both function values negative?

#### Tip:  
Ensure that your initial guesses bracket the root, meaning the function values should have opposite signs to guarantee convergence.

Learn how to use the False Position (Regula Falsi) method to find a positive root of the equation x^3 - 4x + 1 = 0. Step-by-step solution with initial guesses, calculation of approximations, and iterative refinement. Suitable for advanced learners and those interested in numerical methods.

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