Since the process of finding derivatives, and to some extent solving equations, is a mechanical process, it should not be surprising that computers can help. Although you should be able to do most calculations by hand, you may want to use a computer to do some of the work. This will allow you to test that you understand the procedures without performing so much calculuation. In this example, we will use the computer algebra system Sage to find and test critical points. Sage can also graph functions, which allows us to check if our conclusions appear to be correct. Note that graphs produced by a computer must be viewed with an informed eye. Depending on the scale of the graph, some features may not be visible, and of course we can only show a limited portion of the graph.

First, define a function, take the derivative, and display it.

Now find the roots.

Check the concavity at these points. First, the second derivative.

The derivative is zero at 0, 1, and 3, so we check those points.

There is a local maximum at 1, a local minimum at 3, and we can't tell what happens at 0. Now we graph it:

This is not very useful because of the scale. Let's shrink the range.

Now we can see that indeed we have correctly identified the local maximum and minimum. Moreover, it seems quite clear that there is neither a maximum nor a minimum at 0. We can perform the first derivative test to verify this.

Since the derivative is positive in both cases, there indeed is no maximum or minimum at 0. We might also check the concavity:

So we see there is an inflection point at 0.