Suppose that $f:\mathbb{R}_+ \mapsto [0,1]$ is continuous, twice differentiable and non-decreasing, i.e., $f'(x) \ge 0$. For what follows, $f$ could be thought of as a distribution function, if it is helpful, that is, $f(0)=0$ and $\lim_{x \to \infty} f(x) = 1$. In general, $f$ is not assumed to be either convex or concave.
For any fixed parameter $ t \ge x$, are there any sufficient conditions on $f$ that ensure $f(x) + f(t-x)$ is convex or concave, other than any test that relies on the second derivative? More explicitly, under what conditions can $t/2$ be a global minimum (if the sum is convex) or that either $0$ or $t$ can be a global minimum (if the sum is concave), other than, e.g., $f''(t/2) \ge 0$?