We return to the general case, $f$ being any function of class $\mathcal { C } ^ { 3 }$. We assume that $f$ vanishes at a point $x ^ { * } \in I$, for which $f ^ { \prime } \left( x ^ { * } \right) > 0$.
(a) Show that there exists $\epsilon > 0$ such that $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] \subset I$ and $f ^ { \prime } > 0$ on the interval $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. We fix such an $\epsilon$ for the rest and we define $$M = \sup _ { ( x , y ) \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] ^ { 2 } } \left| \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( x , y ) \right| .$$
(b) We assume that $x _ { n - 1 } , x _ { n } \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. Show that $$\left| x _ { n + 1 } - x ^ { * } \right| \leqslant M \left| x _ { n - 1 } - x ^ { * } \right| \cdot \left| x _ { n } - x ^ { * } \right| .$$
(c) We fix $\left. \epsilon ^ { \prime } \in \right] 0 , \epsilon ]$ such that $M \epsilon ^ { \prime } < 1$. Show that if $x _ { 0 } , x _ { 1 }$ belong to $\left[ x ^ { * } - \epsilon ^ { \prime } , x ^ { * } + \epsilon ^ { \prime } \right]$ then the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined and converges to $x ^ { * }$.