Newton's method question

Question

Use Newton's method to approximate a root of the equation ³ as follows.

Let ¹ be the initial approximation.

The second approximation ² is

third approximation ³ is 

Answer 1

f(x) = x³ + x+ 2; f'(x) = 3x² + 1

x_{n + 1} = x_n - f(x_n)/f'(x_n), n ≥ 1

x₁ = - 1

x₂ = -1

x₃= -1.

This is because x = -1 is exact zero of the equation