Definition: Let $\alpha: I \mapsto \mathbb{R^3}$ and $\beta: \overline{I} \mapsto \mathbb{R^3}$ be two regular curves such that $\alpha(t_0) = \beta(t_0)$, where $t_0 \in I \cap \overline{I}$. $\alpha$ and $\beta$ are said to have contact of order $n$ at $t_0$ ($n$ being an integer $\geq 1$) if all derivatives of order $\leq n$ of the functions $\alpha$ and $\beta$ are equal at $t_0$ and derivatives of order $n + 1$ are different.
Let $\alpha(t) = (t, y(t), 0)$ and $\overline{\alpha}(t) = (t, \overline{y}(t), 0)$, $t \in I$ be two regular curves with order of contact $n$ at $t_0 \in I$. Prove that:
a) If $n$ is odd, there is a neighborhood $J \subset I$ of $t_0$ such that $\forall t \in J$, $y(t) - \overline{y}(t)$ doesn't change sign, as shown below.
b) If $n$ is even, there exist $t_1$ and $t_2 \in I$ such that $y(t_1) - \overline{y}(t_1) < 0$ and $y(t_2) - \overline{y}(t_2) > 0$, as shown below.
To be honest I don't have a clue how to go about solving this. My only strategies were expanding $y$ and $\overline{y}$ in terms of their Taylor series and see if there were any insights there... alas, there weren't. I'd appreciate any help. How do I start solving this?