Why does Newton's Method Diverge when f''(r)=0 or r is an inflection point?

Question

I understand that when f'(x) approaches zero near the root Newton's Method will most likely not work because f(x)/f'(x) will shoot off to a large value causing divergence. But how come f''(x) also cannot be near zero at the vicinity of the root? What does a zero second derivative have to do with Newton's Method?

Answer 1

The first example that comes to mind is $\sin x$ in radians. Take the $A = \arctan 2.$ We get $\cos A = \frac{1}{\sqrt 5}$ and $\sin A = \frac{2}{\sqrt 5}.$

If you start Newton's method at $x=A,$ the next stop is $x=-A,$ third $x=A,$ fourth $x = -A,$ forever.