Why does the newton's method not always work well. Also what is a specific function example I could use to support my explanation?
My professor wants us to read the subject on our own and then do a project on it! I'm bad enough at math as it is. Please help me understand what this is all about.

There can be some problems with Newton's method, but overall, it works well. Especially, for solving hard to solve equations, such as

x^{3}+x-1=0

But, some reasons it does not work is suppsoe we have f'(x_{n})=0 for some n, then Newton involves a division by zero, making it impossible to generate x_{n+1}.

But, this is to be expected because the tangent line to y=f(x) is parallel to the x-axis where f'(x_{n})=0, and so the tangent line does not cross the x axis to generate the next approximation.

Sometimes the value of x_{n} produced by Newton do not converge to a solution. Take for example,

x^{\frac{1}{3}}=0 which has x=0 as its only solution, and try to approximate by using Newton with a starting value of x_{0}=1.
If we do this, then we quickly see that the values do not converge to 0 and Newton does not work.

Here is a graph. Note how the lines do not converge around 0.

At the bottom is the graph for one that does converge, x^{3}+x-1=0

Note how the tangent lines are grouping around the solution, which is about 59/86