It makes the light diverge.

correct.

If you can differentiate the function, then you can tell that the graph is concave down if the second derivative is negative over the range examined. As an example: for f(x) = -x2, f'(x) = -2x and f"(x) = -2 < 0, so the function will be everywhere concave down.

If the sign of the term involving x2 is positive then it is concave upward, provided everybody agrees what concave upwards means. Just like y=x2 is concave upwards.

Concave mirrors focus light inward to one focal point. They are often used in telescopes and to magnify the face in makeup mirrors.