Function is bounded from below if sum of partial derivatives is positive

Question

Let $f:\mathbf{R}^n \to \mathbf{R}$ be differentiable, $\sum_{i=1}^n y_i \frac{\partial f}{\partial x_i}(y)\geq 0$ for all $y=(y_1,...,y_n)\in \mathbf{R}^n$. How do I show that $f$ is bounded from below by $f(0)$?

Answer 1

Hint:

The expression from your question is the derivative of f with respect to the radius squared. So if that is positive, then...

Answer 2

Isn't $$ f:\mathbb{R}^2 \rightarrow \mathbb{R}: (x_1,x_2) \mapsto x_1 + x_2 $$ a counterexample?