For my Master's thesis, I'd like to prove the following (but I'm not sure it's true):

On a two-dimensional Riemannian manifold (oriented and closed), for any smooth function $f$, it holds that $$ \int_M \left( 2 |\nabla^2 f |^2 + \text{Scal} |\nabla f|^2 \right) \text{d} V \geq 0, $$ where Scal denotes the scalar curvature. It feel like I have to use some divergence/integration by parts theorem and use that $M$ is Einstein but I just can't make it. Thanks in advance!