# Which conditions should the function follow to be bounded from below

Let us consider that $$f:\mathbb{R}\rightarrow\mathbb{R}$$ is a smooth function and $$A, B$$ are positive constants.

Let us define the function $$g(x)$$ given by: $$g(x) = -A f(x) + B f''(x).$$

I wonder about conditions of $$f(x)$$ that makes $$g(x)$$ bounded from below.

For example if $$f(x)$$ is bounded from above and convex then of course $$g(x)$$ is bounded from below. But I want to find more general rules.