Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for all $k$ and converges to $0$.
Where these constants are as in this theorem:
$f'\in Lip_{\gamma}(D)$ means for any $x,y \in D$, $|f'(x)-f'(y)|\leq \gamma |x-y|$. First we notice that $x^*=0$ is the root we need the method to converge for. What I am thinking of is that since $f''(x)>\delta? 0$, that means $f$ is concave up, also $f'(x)>\rho>0$ implies $f$ is increasing and both implies $f$ is 1-1.
Please help me solve this problem.