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 $t \gt 0$ ?