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3 edited body

This is a job for the Morse lemma. The second degree Taylor polynomial of $F(x,y)$ has the form $ax^2+by^2$ where $ab<0$. (You said $>0$ but that can't be what you meant.) The Morse Lemma says, for a sufficiently smooth function of several variables, that if it has zero constant and linear parts and a nondegenerate quadratic part then there is a change of variables making the function purely quadratic. In the case at hand (two variables and indefinite quadratic part) that means that $F=uv$ where $u(x,y)$ and $v(x,y)$ have no constant term and have linearly independent linear parts whose product is $ax^2+by^2$. Now the implicit function theorem applies to $u$ and to $v$ and you find that the solution set of $F(x,y)=0$ is locally the union of the graphs of two smooth functions, one with positive slope and one with negative slope.

EDIT Here is a proof of the Morse Lemma:

First, in one variable it says that if $f(0)=f_x(0)=0$ and $f_{xx}(0)\neq 0$ then in a neighborhood of $x=0$ $f(x)$ is plus or minus the square of a smooth function. This is true because the vanishing of $f(0)$ means that $f(x)=xg(x)$ for some smooth $g$, and the vanishing of $f_x(0)=g(0)$ means that $g(x)=xh(x)$ for some smooth $h$, and the nonvanishing of $f_{xx}(0)=2g_x(0)=2h(0)$ means that $h$ is locally plus or minus a square.

Now suppose that $F(x,y)$ is such that $F$, $F_x$, and $F_y$ vanish at $x=y=0$ but $F_{yy}$ does not, and suppose that the homogeneous quadratic part of the Taylor series is nondegenerate. The implicit function theorem applied to $F_y$ shows that $F_y=0$ along the graph of a smooth function $y=k(x)$ with $k(0)=0$. Define $f(x)=F(x,k(x))$. Then $F(x,y)-f(x)$ is such that both it and its $y$-derivative vanish along that graph. Therefore we may write $F(x,y)-f(x)=(y-g(x))^2H(x,y)$ F(x,y)-f(x)=(y-k(x))^2H(x,y)$for some smooth$H$. Furthermore$2H(0,0)=F_{yy}(0,0)\neq 0$, so$H$is plus or minus the square of a smooth function. Now$F(x,y)$is the sum of$F(x,y)-f(x)$, which is the square of a smooth function, and$f(x)$, which must also be plus or minus the square of a smooth function by the one-variable case. And so on. 2 added 2137 characters in body; deleted 841 characters in body EDIT Here is a proof of the Morse Lemma: First, in one variable it says that if$f(0)=f_x(0)=0$and$f_{xx}(0)\neq 0$then in a neighborhood of$x=0f(x)$is plus or minus the square of a smooth function. This is true because the vanishing of$f(0)$means that$f(x)=xg(x)$for some smooth$g$, and the vanishing of$f_x(0)=g(0)$means that$g(x)=xh(x)$for some smooth$h$, and the nonvanishing of$f_{xx}(0)=2g_x(0)=2h(0)$means that$h$is locally plus or minus a square. Now suppose that$F(x,y)$is such that$F$,$F_x$, and$F_y$vanish at$x=y=0$but$F_{yy}$does not, and suppose that the homogeneous quadratic part of the Taylor series is nondegenerate. The implicit function theorem applied to$F_y$shows that$F_y=0$along the graph of a smooth function$y=k(x)$with$k(0)=0$. Define$f(x)=F(x,k(x))$. Then$F(x,y)-f(x)$is such that both it and its$y$-derivative vanish along that graph. Therefore we may write$F(x,y)-f(x)=(y-g(x))^2H(x,y)$for some smooth$H$. Furthermore$2H(0,0)=F_{yy}(0,0)\neq 0$, so$H$is plus or minus the square of a smooth function. Now$F(x,y)$is the sum of$F(x,y)-f(x)$, which is the square of a smooth function, and$f(x)$, which must also be plus or minus the square of a smooth function by the one-variable case. And so on. 1 This is a job for the Morse lemma. The second degree Taylor polynomial of$F(x,y)$has the form$ax^2+by^2$where$ab<0$. (You said$>0$but that can't be what you meant.) The Morse Lemma says, for a sufficiently smooth function of several variables, that if it has zero constant and linear parts and a nondegenerate quadratic part then there is a change of variables making the function purely quadratic. In the case at hand (two variables and indefinite quadratic part) that means that$F=uv$where$u(x,y)$and$v(x,y)$have no constant term and have linearly independent linear parts whose product is$ax^2+by^2$. Now the implicit function theorem applies to$u$and to$v$and you find that the solution set of$F(x,y)=0\$ is locally the union of the graphs of two smooth functions, one with positive slope and one with negative slope.