Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 18 / b / Solution

For the specified polynomial,The first equation $F_y=0$ forces $y=x^2$; substitution into $F_x=0$ then gives $x=0$, and hence $y=0$. It also satisfies the curve equation. Therefore the unique singular point is