I have a function $g(x)=-e^{-x^2}$. The task is to use a Newton’s method on the function $g'$ and find a value $\alpha$ such that if $x_0 \in [0,\alpha)$ the Newton's method converges to $0$ and if $x_0>\alpha$ the Newton's method diverges.
I was thinking about $\alpha=\frac{\sqrt2}{2}$ but I can't show that it is divergent and convergent for $x_0$ in particular range. Can someone help me with that?
