Answer:
[tex]S=\{-0.45339,...\}[/tex]
Step-by-step explanation:
Hi!
Greetings
1) Firstly let's graph the function using an utility (Check below). That's how we're going to pick our [tex]x_1[/tex]
2) Let's now use the algorithm. Since we've seen the graph let's now choose one approximation to start with:
[tex]x_{n+1}=x-\frac{f(x)}{f'(x)}\\x_{1}=-0.4\\x_{2}=(-0.4)-\frac{(-0.4)^{3}+2(-0.4)+1}{3(-0.4)^2+2} \approx -0.45483\\x_{3}=(-0.45483)-\frac{(-0.45483)^{3}+2(-0.45483)+1}{3(-0.45483 )^2+2}\approx -0.45339\\x_{4}=(-0.45339)-\frac{(-0.45339)^{3}+2(-0.45339)+1}{3(-0.45339 )^2+2}\approx -0.45339\\[/tex]
Since the approximations started to repeat, then it is safe to say that
this approximation is one of the roots for the function [tex]f(x)=x^{3}+2x+1[/tex]
