Newton-Raphson Method and its Application

Introduction:
The Newton-Raphson method is an iterative technique used to find the real root of a given equation. It is widely used in various fields, including engineering, physics, and mathematics. The method involves making an initial guess, denoted as x0, and then using the formula:

x1 = x0 - f(x0)/f'(x0)

where f(x) is the given equation and f'(x) represents the derivative of the equation with respect to x.

Given Equation:
The given equation is x^3 - 3x^2 + 3x - 5 = 0. We need to find the number of different values of x0 for which the Newton-Raphson method fails for this equation.

Explanation:
To understand why the Newton-Raphson method fails for certain values of x0, we need to consider the behavior of the method when applied to this particular equation.

1. Define the Function and its Derivative:
- f(x) = x^3 - 3x^2 + 3x - 5
- f'(x) = 3x^2 - 6x + 3

2. Find the Roots of the Equation:
- By applying the Newton-Raphson method, we can find the roots of the equation. Let's denote the root as x*.

3. Analyze the Convergence of the Method:
- For the Newton-Raphson method to converge, the initial guess x0 must be reasonably close to the root x*.
- The convergence is determined by the behavior of the derivative f'(x) near the root.
- If f'(x) is close to zero or becomes zero at any point, the method fails to converge.

4. Calculate the Derivative at different points:
- By substituting the values of x in f'(x), we can observe the behavior of the derivative.
- For example, let's substitute x = 0, x = 1, and x = 2 in f'(x).

- f'(0) = 3(0)^2 - 6(0) + 3 = 3
- f'(1) = 3(1)^2 - 6(1) + 3 = 0
- f'(2) = 3(2)^2 - 6(2) + 3 = 3

5. Evaluate the Convergence:
- From the above calculations, we can see that the derivative f'(x) becomes zero at x = 1.
- This indicates that the Newton-Raphson method fails to converge for the initial guess x0 = 1, as the derivative becomes zero and the method cannot proceed further.

Conclusion:
In conclusion, the Newton-Raphson method fails for one specific value of x0, which is x0 = 1, because the derivative of the given equation becomes zero at x = 1. For all other initial guesses, the method should converge and find the real root of the equation.

Newton-Raphson method fails if f'(x) = 0