Problem

Show that the newton method applied to $r(x) = -x^5 + x^3 + 4x$ where the inital point is $x_0 = 1$ produces a cyclic behavior

Solution

Step:

1) Find the roots of $-x^5 + x^3 + 4x$ $$x(-x^4 + x^2 + 4)$$


$$-x(x^4 - x^2 - 4)$$


$$-x (y^2 - y - 4) | y = x^2$$

2) Root of (y^2 - y - 4) = $\frac{1}{2} \pm \sqrt{17} $

3) Root of -x = 0

Since these roots are distinct and non degenerate, they will never converge to a single point and rather jump between these values.