Consider the polynomial equation $x^{5}-x-1=0$
(a) Show that this equation has a solution x = r with $1 <r <2$
(b) Define an appropriate contractive sequence $X_{n}$ and approximate r with an accuracy of $10^{-4}$
I thought about rewriting the equation as $x=x^{5}-1$ but it didn't work because the succession that resulted was not contractive. How can I solve the problem? I think we can use the mean value theorem of differential functions, if they could help me it would be good. Thank you
Hints:
(a) Use the intermediate value theorem.
(b) Use Newton's method with $x_0\ge2$.