ODEs - Stability

We have the system of ODEs

$y_1'=y_2-y_1 \cdot f((y_1,y_2)^T)$

$y_2'=-y_1-y_2 \cdot f((y_1,y_2)^T)$,

where $f \in C^1(\mathbb{R}^2, \mathbb{R})$.

Prove the following statement:

If $f((y_1,y_2)^T)>0$ in a neighbourhood of $(0,0)$, then $(0,0)$ is asymptotically stable, but if $f((y_1,y_2)^T)<0$ in a neighbourhood of $(0,0)$, then $(0,0)$ is instable.

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.