Suppose $F(x,y)$ is a smooth function. The linear term of its Taylor series about $(a,b)$ is $$F_x(a,b)(x-a)+F_y(a,b)(y-b)$$ which also happens to describe the tangent line at $(a,b)$ to the curve implicitly given by $F(x,y)=0$, provided one of the partials does not vanish.
However, it may happen that both the partials vanish at that particular point and so no tangent line is obtained. But what I noticed is that for polynomial functions passing through the origin, the lowest degree term is the product of 'tangent-like' lines to the curve drawn at the origin. My questions are :
- Why is this happening for polynomial functions?
- Does this happen for general smooth functions?
I mentioned the first term in the Taylor series since it would be the lowest degree term (which I observed to be the case for polynomial functions).