In middle school, the graph of a quadratic function stretched endlessly, so the maximum and minimum always came from the vertex. But when the range of x is restricted to α≤x≤β, things change — depending on whether the vertex lies inside or outside the range, the locations of the maximum and minimum can change. Move the range with the sliders and see for yourself.
Solution strategy: There are always only three candidates — the function value at the vertex (only when the vertex is inside the range), and the endpoint values f(α) and f(β). The largest of these is the maximum, and the smallest is the minimum. Because the values you need to calculate are clearly determined, drawing the graph and checking which points are higher or lower can help prevent mistakes.
When the vertex is outside the range, the function simply keeps increasing or keeps decreasing throughout that interval. In that case, there are only two candidates, the two endpoints, so the comparison is even simpler.