Define the gradient $\nabla f(x(t))$ as the vector with components $\partial f/\partial x_i$. If $x$ is a function of $t$ with derivative x'(t)=v(t), how can I show that
$ v \cdot \nabla f(x(t)) \geq 0 $
when $t=0$ if $f(x(0))=0$
and if $f(x(t))\geq0$ when $t \gt 0$ ?