The Derivative in Function Analysis - Critical Points

Let's suppose that you have the function
Take its derivative. Now, let's plot their graphs.

As you see, the derivative is negative for all negatives, and positive for all positives.
So, in proper terms, is monotonically decreasing, actually, strictly decreasing over the interval
It's also monotonically increasing, strictly so, over the interval .
But what happens at 0?

As you obviously can see, the graph of is a parabola. At zero, it has its "lowest" point, a minimum.

But then after zero, it starts going back up again. It goes from decreasing to increasing.
Around this point, the derivative changes sign.

It can also happen for, say, a maximum. It can be going up and then start going back down again.
If this is the case, and this point is a maximum or minimum of a function continuous at that point, and the derivative switches sign around that point, then this point is a critical point.