Q:

How do we know when two curves touch each other?

What are the conditions of two curves touching each other?
A necessary condition for this is that the derivative for both the curves should be the same at the point of intersection.
But that doesn't seem to be sufficient, as in the case of $y = x^3$ and $y=x^5$, both of which have the same derivative at x = 0, but they are intersecting or crossing each other.
Does it have something to do with point of inflection? I'm unable to arrive at the conditions precisely.

A:

Curves may touch and cross, nonetheless. Touching is defined as having the same tangent vector in a common point. If you are looking at graphs of functions and want one resulting curve to be completely on on side of the other one locally, the lowest order derivitave which is not the same for both functions has to be larger than the corresponding derivative of the other function in a neighbourhood of the common point. Then Taylors theorem implies that one has to lie completely on one side of the other one. This assumes the existence of suffiently high order derivatives, of course.

