Definitions of offensively:

• adverb:   in an aggressive manner

  Example: "`In this crisis, we must act offensively,' the President said"

• adverb:   in an obnoxious manner

  Example: "He said so in one of his more offensively intellectually arrogant sentences"

• adverb:   in an unpleasantly offensive manner

  Example: "He smelled offensively unwashed"