The box that must be opened is the one labeled “apples and oranges.”  By definition, whichever fruit is inside, is the only fruit type that that box contains.  Let’s say that you found an apple in that box that was labeled with both apples and oranges; because you know it must therefore only contain apples, then you conclude that the box that is labeled “oranges” cannot contain only oranges, as all boxes have been said to be mislabeled.  Thus, the box labeled “oranges” must contain both apples and oranges, leaving the box labeled “apples” to contain only oranges.

The question is faulty.

The question says: “boxes have been incorrectly labeled such that no label identifies the actual contents of the box it labels”. An ‘incorrect’ label on a box NEED NOT necessarily match any other box (e.g. Box containing apples bearing label which says ‘potatoes’, the box of oranges having label that says ‘pears’ and the box containing apples and oranges labelled ‘grapes’ , are all ‘incorrectly labeled‘, satisfying the condition mentioned in the puzzle. If this be the case, the only way you can correct the labels is by checking ALL boxes.)