I think what this test wants is for you to say "positively," but that still sounds quite wrong to me (this is not how we use the word positively even if it has a meaning which seems to fit the context) and none of the four options sound good to me as a native speaker or as a teacher. I would have written "definitively." There are also multiple errors in the English of this test: • "For each of the empty space" should have "spaces" • "overweight" is an adjective, not a noun. It should be "obesity" here. This is not a good test, as far as I can tell.

I agree with Eric (below) that 'definitively' works best, but out of the options you have I'd say 'categorically' would be my choice.