Let $f:[1, +\infty)\rightarrow\mathbb{R}$ be a concave function. Suppose $F:[1, +\infty)\rightarrow\mathbb{R}$ is a primitive function of $f$. My questions are:
(a) What other condition is required to monotone $f$?
(b) Is $F$ concave?
I know that, suppose $f$ is differentiable on $(a, b)$. Then $f$ is concave if and only if $f^\prime$ is decreasing. Also, $f$ is concave, then for any $x$ and $y$ in the interval and for any $\alpha \in [0,1]$ $f(\alpha x + (1 - \alpha)y) \geq \alpha f(x) + (1 - \alpha)f(y)$.
For a), you can use something along those lines:
$$\lim_{x \to -\infty} f^\prime(x) < 0$$
Having that and concavity $f^{\prime\prime}(x) < 0$, you can easily conclude that $f^\prime(x) < 0$, so f is a decreasing concave function.
There's also the symmetric case, which yields an increasing concave function: $$\lim_{x \to \infty} f^\prime(x) > 0$$