There is a difference between aninflection pointand anextremum.

Don't quote me on the exact definitions, but I can tell you the general idea. Anextremumis when a function has a maximum or minimum value. This happens because the derivativechanges 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.

Aninflection pointis when thesecond 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.