Suppose this is $f'(x)$:
And I am being asked if $f(x)$ has global extremum.
Since the endpoints are open, there can't be a global extremum there. So now I only have to check for B, D and F.
But without any numbers, I feel like I can't really check verify those points.
For example, B and E are clearly a local minimums, but how can I tell which one is "lower" in $f(x)$? And even more importantly, even if I figure that out, how could I verify that it really is the lowest $f(x)$ can be?
D is the only local maximum so at least I don't need to compare it with any other critical point, but I still can't guarantee that $f(D)$ is the highest it can get.
I was thinking that I could reason that the "area inside the curve" is larger/smaller at certain intervals so that would allow me to compare critical points, but that's related to integration I think and beyond the scope of my homework, so I feel there is something more fundamental that I am missing here.
For this, we can use the first derivative test. Essentially, if the graph of $f'(x)$ goes from positive to negative at a point, that point is a maximum. Points B and F go from negative to positive, so they can't be maximums. Point D, however, goes from positive to negative. Therefore, point D is your global maximum.