So this question is directly related to a comment made by David Mumford in his
Lecture 1 given at U. Michigan in 1974 entitled: *What is a curve and how explicitly can we describe them ?*

Mumford claims that if you take a (non-hyper elliptic) smooth projective curve $C$ over $\mathbb{C}$ of genus 3 and embed it in $\mathbf{P}^2$ via its canonical map (denoting the image of the curve again by $C$), then there are exactly $108$ points $x\in C$ for which there is a conic passing through $x$ with contact order (with respect to $C$) equal to $6$.

Q1: How does one prove that you have only finitely many such conics touching $C$ and have contact order $6$?

This seems to suggest, that for most points $P\in C$, the best contact order of a conic passing through $P$ is $5$.

Q2: In general if $C\subseteq \mathbf{P}^2$ is a fixed embedded smooth projective curve and $x\in C$ is a point, then for a fixed degree $d$, how does one compute the maximum contact order at $x$ among all smooth projective curves $D$ of degree $d$ in $\mathbf{P}^2$ passing through $x$ (is this computable)?