Critical points of autonomous ODEs

Question

I am reading a book which makes the following claim:

Suppose $f:\mathbb{R}^n\to\mathbb{R}^n$ is $C^1$ function and consider the autonomous system of ODEs $$ \nabla{x}(t)= f(x(t)). $$ Let $x(t)$ be a (local) solution to the above system with initial state $x_0$.

If $f(x_0)\equiv 0$, then $x_0$ is a stationary point of our local dynamical system.

While I suppose the statement is intuitive, I am having a hard time proving it rigorously. Any help is appreciated!

Note: By "$x_0$ is a stationary point", I mean that if $x(0) = x_0$, then $x(t) = x_0$ for all admissible $t$.

Answer 1

Just plug $x\left(t\right)=x_{0}$ into the equation and note that it is satisfied. Because the solution is unique, if $x\left(0\right)=x_{0}$ this must be the solution at all times.