$f:\mathbf{R} \rightarrow \mathbf{R}$ is twice differentiable. $f''(x) \leq 0$ $\forall x \in \mathbf{R}$. $f$ is also bounded below. Show $f$ is a constant function.
I've got to $f(x+y)-f(x) \leq f(x)-f(x-y)\, \forall x \in \mathbf{R}, y>0$ using Rolle's Theorem and I think I'm making an argument about $f'(x)$ being a decreasing function, but I can't see how to get to $f$ constant.