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We use the First Derivative Test to determine if a critical point is a maximum or minimum or neither. The idea is to break up the function into separate sections and then to analyze each section. The function is broken at discontinuities and critical points. So the first step is to make sure we know what the domain is and then to find the critical points. Once we have a set of points that are either discontinuities or critical points, we test values in between the breaks. We plug each test point into the first derivative. If the value is greater than zero, then the function is increasing. If the value is less than zero, then the function is decreasing.

To keep track of what we are doing, we can build a table like this one. Let's say we have a function \(g(x)\) and we have determined break points (critical points and discontinuities) at \(x=c_1\) and \(x=c_2\) and the function is defined for \(x < c_1\) and for \(x > c_2 \). Also, the entire domain needs to be covered by the intervals in the first row.

So, how to do you know what test x-value to use? Basically, you can use any value inside the interval in the first row. Notice that the intervals are open, meaning that they do not include the endpoints. So the point you choose cannot be an endpoint but can be any other point in the interval.

The possible conclusions for the interval columns are increasing or decreasing. The possible conclusions for the point columns are maximum or minimum or neither, if the point is in the domain of \(g(x)\). If the point is not in the domain, we just leave it blank since it has no meaning.

A critical point is a minimum if the function goes from decreasing to increasing, left to right.
A critical point is a maximum if the function goes from increasing to decreasing, left to right.
If the function does not change direction, the critical point is neither a maximum nor a minimum, which we call a saddle point.

A maximum or minimum can be referred to as an extremum ( plural extrema ) if our discussion can apply to either one.
If the extremum is the largest (for a maximum) or the smallest (for a minimum) value of the function everywhere, it is called an absolute extremum. Otherwise, it is called a relative extremum. Relative extremum are sometimes called local extremum.

Before going on to practice problems, here is a good video to watch that explains all of this with an example.

Okay, time for some practice problems and then we start looking at the second derivative and what it can tell us about the graph.