I'm trying to solve the following problem:
Assume $f$ is a $C^1$ vector field on $\mathbb{R}^n$ and $f(0) = 0$. Suppose some eigenvalue of $Df(0)$ has positive real part. Show that in every neighborhood of $0$ there is a solution $x(t)$ to the equation $\dot{x} = f(x)$ for which $|x(t)|$ is increasing on some interval $[0,t_0]$, $t_0 > 0$.
I know that it has something to do with the linearization $\dot{v} = Av$, where $A = Df(0)$, and the fact that it has a unstable manifold. I want some help to write it corretly and formaly.