Choice A is the correct answer. The direction a parabola opens is determined by the leading coefficient $a$.

1. Here, $f(x) = -1x^2 + 6x$, so $a = -1$.
2. Since $a < 0$, the parabola opens downward (like a frown).
3. Therefore, the vertex is the highest point on the graph, representing a Maximum.

Choice B is incorrect because a minimum occurs when $a > 0$ (opens upward). Choice C is incorrect because a parabola cannot have both. Choice D is incorrect because every parabola has a vertex that is either a max or min.