If f (x0) = 0, the test fails and one has to investigate further, by taking more derivatives, or getting more information about the graph. Besides being a maximum or minimum, such a point could also be a horizontal point of inflection.

Can a second derivative not exist?

In order for the second derivative to change signs, it must either be zero or be undefined. So to find the inflection points of a function we only need to check the points where f ”(x) is 0 or undefined. Note that it is not enough for the second derivative to be zero or undefined.

What do you do when the second derivative test is 0?

Since the second derivative is zero, the function is neither concave up nor concave down at x = 0. It could be still be a local maximum or a local minimum and it even could be an inflection point.