Consider system of nonlinear equations: \begin{align} 2x+xy-y &= a \\ x+4y-y^3+x^2 &=b \end{align} Using the inverse function theorem, show that the function has a unique solution for sufficiently small $a$ and $b$.
Attempt: I have not gone through the implicit function theorem yet, so here goes. We wish to show that the Jacobian determinant is nonzero at points $(x_0,y_0)$ such that $F(x_0,y_0)=(a,b)$, then by the inverse function theorem, it is one-to-one.
Computing the Jacobian determinant, $$ \det \left( \begin{bmatrix} 2+y & x-1 \\ 1+2x & 4-3y^2 \end{bmatrix} \right) =-3y^3-6y^2+4y+1-2x^2 + x + 9 $$ I'm not really sure where to go from here, and how to find $(x_0,y_0);F(x_0,y_0)=(a,b)$ or quite frankly if this is the correct approach. Any input would be greatly appriciated!