  We prove that Viro's patchworking produces real algebraic curves with the
maximal number of real inflection points. In particular this implies that
maximally inflected real algebraic $M$-curves realize many isotopy types. The
strategy we adopt in this paper is tropical: we study tropical limits of
inflection points of classical plane algebraic curves. The main tropical tool
we use to understand these tropical inflection points are tropical
modifications.
