Question:F,G are functions $\mathbb R^2 \to \mathbb R^2$,where $(a,b)=F(x,y)=(x^3-y^3,x^2+2xy^2)$,$(u,v)=G(a,b)=(a^5+b,b^5-a)$,can x, y be defined as continuously differentiable functions of u, v in a neighborhood of $(1,0)$?
My thoughts: Since it is easy to show that $G \circ F$ is continuously differentiable,I want to find a $(x,y)$ such that $G(F(x,y))=(1,0)$ and check whether $\det((G \circ F)'(x,y))$ is zero or not.And if it is not zero, then using inverse function theorem can show that there exist a continuously differentiable inverse function in a neighborhood of $(1,0)$. But I am stuck at finding such $(x,y)$ which let $G(F(x,y))=(1,0)$.