= Solution
{c}
For fixed $x,y$, the map $z\mapsto z^3+z+xy-1$ is strictly increasing because its derivative is $3z^2+1>0$, and it tends to opposite infinities at the two ends of the real line. The <intermediate value theorem> gives exactly one <root>, defining $g(x,y)$.
For $F(x,y,z)=z^3+z+xy-1$, $F_z=3z^2+1$ never vanishes. The <implicit function theorem> supplies a $C^1$ solution locally around every $(x,y)$. Uniqueness makes all these local functions agree on overlaps, proving $g\in C^1(\mathbb R^2)$. Implicit differentiation gives
$$
Dg(x,y)=-\frac{(y,x)}{3g(x,y)^2+1}.
$$
At $(1,1)$ the <root> is zero, so \b[$Dg(1,1)=(-1,-1)$].
Back to article page
