Stability of zero

Question

Determine stability of zero in \begin{cases} x'=y \\ y'=-f(x) \end{cases}

Here $f: \mathbb{R} \rightarrow \mathbb{R}$ is class $\mathcal{C}^1, f(0)=0$ and $xf(x)>0$ for $x \neq 0$.

Could you help me solve this problem?

Answer 1

Hint: Find a Lyapunov function $V$ of the form $$ V(x,y)=F(x)+y^2, $$ for some suitable nonnegative function $F$.