Number of inflection points of real planar cubic curve

How many inflection points can a cubic planar curve given by $F(x,y)=0$, where $\deg F=3$, $x,y\in\mathbb{R}$, have? I know that inflection points correspond to the roots of $F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x$, but I am unaware of any results on roots of systems of polynomial equations over $\mathbb{R}$. I encountered this problem in a differential geometry course at IUM. I have seen some solutions of similar questions, but all of them involved algebraic geometry and gave answers only over an algebraically closed field and in a projective space.

Solution 1:

It is a standard fact (look at any textbook of Plane Algebraic Curves) that the number of inflection points of an irreducible complex projective cubic is:

• 9 if the cubic is smooth
• 3 if the cubic has a node
• 1 if the cubic has a cusp

Moreover, they have the property that, any time you join two inflection points with a line, the third point in which the line meets the cubic is also an inflection point. This property prevents the cubic from having more than 3 inflection points with real coordinates.