Let’s F be a function of two variables (x,y) which is continuous in a neighborhood of (x0,y0) while F(x0,y0) = 0 and suppose it would have continuous partial derivatives Fx ,Fy ,Fxx ,Fxy ,Fyy,Fyx in this neighborhood and Fy(x0,y0) # 0 .

Prove that F(x,y) has a unique answer as function f in a neighborhood of x0 such that F(x,f(x)) = 0 and f is continuously two times differentiable by the following formula :

f”(x) = - (Fxx(x,y)( Fy(x,y))^2 - 2 Fx(x,y) Fxy(x,y)+ Fyy(x,y)( Fx(x,y))^2 )/ (Fy(x,y))^3