There is a difference between an inflection point and an extremum.
Don't quote me on the exact definitions, but I can tell you the general idea. An extremum is when a function has a maximum or minimum value. This happens because the derivative changes sign, because that in turn means that the function switched from rising to falling or vice versa, creating a "hill." If the function is differentiable, the derivative at an extremum will be 0, but the converse (vice versa) won't necessarily hold, because of the example in your book.
An inflection point is when the second derivative (the derivative of the derivative) changes sign. So if the second derivative exists at that point, it will be equal to 0. This is the point where the concavity of the function changes.