Show that if the Hessian $\nabla ^2 f(\vec{x})\in\mathbb{R}^{nxn}$ is a constant matrix (independent of $\vec{x}$) and a vector $\vec{d}\in\mathbb{R}^n$ is a solution of the system $\vec{0}=\nabla f(\vec{y})+\nabla ^2 f(\vec{y})\vec{d}.\quad$ Then $\nabla f(\vec{y}+\vec{d})=\vec{0}$
My try:
I'm trying to prove it by useing the Taylor's expansion $$\nabla f(\vec{y}+\vec{d})=\nabla f(\vec{y}) + \int_0^1 \nabla ^2f(\vec{y}+t\vec{d})\vec{d}dt= \nabla f(\vec{y}) + \nabla ^2f(\vec{y}+t\vec{d})\vec{d}$$
I'm not sure how to solve the integral explicitly, I don't know if there is a property. Any suggestions would be great!