The sign of the tells you if the function is increasing or decreasing in the neighborhood of . Note that on left of the relative maximum must be an increasing function, and on the right of the relative maximum it must be a decreasing function. It is the other way around in the neighborhood of a relative minimum.
And bear in mind that what tells you about , tells you about .
Solution to the is the candidate for the inflection point, and it actually is an inflection point if on the left and on the right of it is of the opposite sign.