ewton's Method will converge to a true solution if you have a good initial oproximation. If you don't it may not converge at all. Consider, for example, the quation

$f(x)=x(x-1)(x+1)=x^3-x=0.$

bviously, the solutions are

$x=,-1,0,1$

we start Newton's Method with $x_0$ being close to one of these solutions we will get onvergence to that solution. On the other hand, note that

$f^{\text{'}}(x)=3x^2-1$

zero when