Suppose a 4th grade curve meets a line in one point with multiplicity 4.
Example: the lemniscate $(x^2 + y^2)^2 = y^2 - x^2$ meets the line $x=y$ when the condition $x^4=0$ holds. This shows that line $x=y$ meets the lemniscate in one point $(0,0)$ with multiplicity 4. The line $x=y$ therefore is tangent to the curve in $(0,0)$ and also therefore $(0,0)$ is a node.
In an (old) article I read that in such a case obviously the tangents must be flexes, i.e. the nodes are also inflexion points.
Can anybody provide this "obvious" argument?