Let $f: \mathbb{R}^n \to \mathbb{R}$ be a class $C^2$ differentiable function defined over all of $\mathbb{R}^n$. Suppose also that
$\Big|\int \cdots \int_{\mathbb{R}^n}fdV\Big|< \infty$
What restrictions on $f$ must be placed so that $f$ and all of its partial derivatives, or at least the first two, are guaranteed to vanish at infinity?
Edit: As Julián pointed out below, requiring the function to vanish more quickly than some other function does not guarantee that it and all of its partial derivatives vanish at infinity. There must be a more strict characterization of these kinds of functions then.
No, that is not enough; $f$ could oscillate wildly. Consider for instance the following example in $\Bbb R$: $$ f(x)=\frac{\sin(x^4)}{x^2+1}. $$