Parabola Function Analysis: Finding Points Where f(x) < 0

To determine where the function $f(x) < 0$, it's given that the parabola intersects the X-axis exactly at point A, the vertex, indicating the function has its maximum (if it opens downwards) or minimum (if it opens upwards) at this point.

Since it intersects (not crosses) the X-axis at one point, this must mean the parabola opens downwards, having its vertex at the X-axis. Thus, it tests negative to the left and right of point A, except for the vertex A itself, where $f(x) \geq 0$.

Here's the solution approach:

• Step 1: Determine parabolic orientation (vertex as max or min point).
• Step 2: Verify negative function values f(x) outside vertex.
• Step 3: Solution: If the parabola opens downwards and A is the only intersection, intervals are $x < A$ and $x > A$.

The analysis shows negative regions surrounding the vertex for downwards opening, consistent with options (b) and (c).

Therefore, the solutions are Answers (b) + (c) are correct.