Past exam of the options II course of the University of Oxford / 2012 / F / F2 / a / iii / Solution

The planar implicit function theorem states that if $F$ is $C^1$ near $(a,b)$, $F(a,b)=0$, and $F_y(a,b)\ne 0$, then in some product neighbourhood the equation $F(x,y)=0$ is equivalent to $y=h(x)$ for a unique $C^1$ function with $h(a)=b$. Moreover,If instead $F_x(a,b)\ne 0$, exchange the coordinates to solve locally for $x$ as a function of $y$.