(10 points)

1. Let $f(x)$ be twice continuously differentiable on $\Re^n$.
   1. Define local and global minimum for $f(x)$.
   2. Define Newton's method for minimization of $f(x)$.
2. Show that the function $f(x)$ defined on $\Re^1$ by
   
   $$f(x)=x^{\frac 43}$$
   
   
   has a unique global minimizer at $x^*$ but that, for any nonzero initial point $x^{(0)}$, the Newton's Method sequence $\left\{x^{(k)}\right\}$ with initial point $x^{(0)}$ for minimizing $f(x)$ diverges.