(Solved): (a) Formally state the implicit function theorem as it applies to the differentiable function: f(x) ...

(a) Formally state the implicit function theorem as it applies to the differentiable function: $f(x)=0,$ where $f:{\mathbb{R}}^2?\mathbb{R}$. (b) Use the theorem to determine whether the expression: $f(x)=x_1^2?2x_1{x}_2+x_1{x}_2^3=2$ defines $x_2$ as a function $x_2:\mathbb{R}?\mathbb{R}$ near $x^0=(2,1)$. If so, find the parametric form of the best affine approximation of this function around the point $x^0$.