What is the approximate value of X4 using Newton's method for f(x) = x^3 - 2x + 2 starting from X1 = -1.0000?

To determine the approximate value of X4 using Newton's method, we start with the given function ( f(x) = x^3 - 2x + 2 ). Newton's method is based on the formula:

[

X_{n+1} = X_n - \frac{f(X_n)}{f'(X_n)}

]

To apply this method, we first need to calculate the derivative of ( f(x) ):

[

f'(x) = 3x^2 - 2

]

We start with the initial guess ( X_1 = -1.0000 ).

1. Calculate ( f(X_1) ) and ( f'(X_1) ):

[

f(-1) = (-1)^3 - 2(-1) + 2 = -1 + 2 + 2 = 3

]

[

f'(-1) = 3(-1)^2 - 2 = 3(1) - 2 = 1

]

2. Update to find ( X_2 ):

[

X_2 = X_1 - \frac{f(X_1)}{f