 Solution

The required condition is $D_{2}F(a,b)\ne 0$. The implicit function theorem then gives a unique local differentiable branch $y=f(x)$ with $f(a)=b$ andThe denominator remains nonzero after shrinking the neighborhood by continuity.

For the specified smooth function,At $(0,0)$ both equal one, so the condition holds and $f^{\prime }(0)=-1$. Differentiate $F(x,f(x))=0$ twice:At zero, $F_{xx}=F_{yy}=0$, $F_{xy}=1$, and $F_y=1$. ConsequentlyThe additional derivatives exist because this particular $F$ is smooth; the general first-derivative theorem only requires the stated continuous first partial derivatives.