Properties of $g$ satisfying $f(x,x)[\nabla_{x}^{2}g(x,y)]_{x=y}+2[\nabla_{x}f(x,y)]_{x=y}\cdot[\nabla_{x}g(x,y)]_{x=y}=0$ for all $f$

Question

Suppose that $g:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ is a function such that:

$$f\left(x,x\right)\left[\nabla_{x}^{2}g\left(x,y\right)\right]_{x=y}+2\left[\nabla_{x}f\left(x,y\right)\right]_{x=y}\cdot\left[\nabla_{x}g\left(x,y\right)\right]_{x=y}=0$$

holds for all $f:\mathbb{R}^{2n}\rightarrow\mathbb{R}$. Here $x$ and $y$ denote points in $\mathbb{R}^{n}$.

The properties that interest me are:

1. Does it follow that $g$ satisfies $\left[\nabla_{x}^{2}g\left(x,y\right)\right]_{x=y}=0$ ?

2. Does it follow that $g$ satisfies $\left[\nabla_{x}g\left(x,y\right)\right]_{x=y}=0$ ?

Note: I am only considering smooth functions $f$ and $g$ here.

Answer 1

Since this is holding for every $f$, let's take $f = 1$. It answers the first question.

Then take $f(x,y) = x$, it answers the second question.