Taylor for gradients, lower bound on integral remainder

Question

from Taylors Theorem I can express the gradient of a function as follows:

$\|\nabla f(x+p)\|=\|\nabla f(x)+\int_0^1\nabla^2 f(x+tp)pdt\|$

Is there some lower bound on the right hand part? E.g. involving just $\nabla^2f(x)$?

Thanks a lot!