Why the newton's method is not working for some functions?

Question

I know that we cannot use the Newton's method for the function which has $f\prime(x)=0$. But I want to know some more detailed proof for it.

Answer 1

The proof that it won't work in that case is trivial - you get an impossible fraction of $\frac{f(x)}{0}$. But if what you're looking for is "why" it won't work, beyond that obvious reason, consider it geometrically - what Newton's method is is following the tangent line to the x-intercept. If the derivative is zero, the tangent line is horizontal; if it's horizontal, there's no x-intercept. If what you're looking for is a detailed listing of when Newton's method will and won't work, there are too many variegated pathological cases for that to exist. Keeping in mind that geometric interpretation, following the tangent line to the x-intercept, should give you a general sense.

Answer 2

The deep reason is that Newton is a "single point" approach which is unaware of the existence of a root ! It can keep iterating even if there is none, and widly wander.

This contrasts with "two points" methods (such as dichotomy, regula falis or the more sophisticated Brent method) that keep track of a change of sign and narrow it down.

Newton is extremely efficient when the first order Taylor approximation holds

$$f(x)=f(y)+f'(y)(x-y)+R(x,y)$$ with a small remainder, i.e. "close" to a root, and very poor otherwise.