Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial derivatives is non-zero.

I am looking for $F$ which satisfies the following condition: There is a point in $P \in \mathbb{R}^n$ such that when we apply implicit function theorem and obtain a function $G$, all the second partial derivatives of $G$ is $0$ at $P$.

I have been trying to find such $F$ but I have not found one yet. If anybody has any leads I would appreciate it!

Clarification: Let $P= (p_1, ..., p_n)$. Since $\nabla F(P) \neq 0$ without loss of generality suppose $\partial F/\partial x_n (P) \neq 0$. Then the implicit function theorem tells us that there exists some function $G$ satisfying $F(x_1, .., x_{n-1}, G(x_1, .., x_{n-1})) = 0$ for $(x_1, .., x_{n-1})$ in some open neighbourhood of $(p_1, .., p_{n-1})$. I would like all the second partial derivatives of $G$ at $(p_1, .., p_{n-1})$ to be $0$.