When using Newton's Method to find a root, how do you know to what to choose as x1?

Question

When using Newton's Method to find a root, how do you know to what to choose as x1? What could cause Newton's Method to be ineffective. Name at least two potential problems?

Answer 1

You can pick any x₁!

1) determine slope at x₁

2) determine equation of line at x₁

3) determine x-intercept of equation in 2)

4) use x-intercept in 3) as x₂

5) repeat steps 1 - 4