Sometimes Newton's method does not work. This exercise illustrates an example.
Use Newton's method to find a root of $p(x) = x^3 - x$, with the starting value $x_0 = \frac{\sqrt{5}}{5}$, and observe what happens with the first few iterations.
For your calculations, use exact arithmetic (not decimals). For example, enter $x_0$ as $\sqrt{5}/5$.
$x_0 = $
$x_1 = $
$x_2 = $
$x_3 = $
$x_4 = $
Newton's method may fail in many different ways. A detailed analysis of Newton's method, and related methods, is a huge subject and well
beyond the scope of our course.
