Let $$f(u,v)=(u^2+u^2v+10v,u+v^3).$$
a) Prove $f$ has inverse $f^{-1}$ in a neighborhood of the point $(1,1)$;
b) Find an aproximate value of $f^{-1}(11.8,2.2)$.
My work:
Note $$f'(u,v)=\begin{pmatrix} 2u+2uv &1\\ u^2+10 &3v^2 \end{pmatrix},$$ then $$\det(f'(u,v))=(2u+uv)(3v^2)-(u^2+10)=6uv^2+3uv^3-u^2-10=0\\\iff6uv^2+3uv^3-u^2=10\\\iff3u(2v^2+v^3-u)=10\\\iff 2v^2+v^3-u=10.$$
Suppose $2v^2+v^3-u\not = 10$, then $$\det(f'(u,v))\not =0,$$ whis implies $f'(u,v)$ is invertible.
Moreover, note $f$ is of class $C^1$ then by inverse theorem function we have: there exists an open set $U=B((1,1),r)$ with $r=1-ε$ such that $f|_U$ is injective and have local inverse.
I don't know how to make the part b). Can someone help me?