Covariant constant field

Question

Let $f:M\rightarrow\mathbb{R}$ be a smooth function and $\chi(M)$ be the set of smooth vector fields on a complete Riemannian manifold $M$. If the covariant derivative of $grad(f)$ is zero, i.e., $$\nabla_X(grad(f))=0\ \forall X\in \chi(M),$$ then can we conclude that $\Delta|grad(f)|^2$ is constant? Please give any suggestion.