How do I solve for $\delta$ in $[r−\delta,r+\delta]$ where Newton's method will surely converge? For example in:
Explain Newton’s method for $$f(x) = x^3+x−2 = 0.$$ Show that Newton’s method converges if $$x_0 \in [1−1/30 , 1+1/30 ]$$ to a limit $L$. Find an error estimate for the error $$e_n = |x_n−L|.$$ (Hint: $x^3 −3x^2 +2 = (x−1)(x^2 −2x−2)$ and $|x^2 − 2x − 2| ≤ 10$ if $0 ≤ x ≤ 2$.)
How was the $1/30$ obtained?