やっと and ついに: after waiting and a culminating outcome
Compare solving a difficult problem with winning a championship and a machine failing, without treating ついに as inherently negative.
Finally after waiting or difficulty
やっと can express finally reaching something after waiting or difficulty. 難しい問題がやっと解けました。 says that a difficult problem was finally solved.
長く待って、やっと順番が来ました。 says that the person’s turn finally came after a long wait. The waiting is explicitly stated.
An outcome at the end of a course of events
ついに means finally or in the end. チームはついに優勝しました。 says that the team finally won the championship.
機械はついに動かなくなりました。 says that the machine eventually stopped working. ついに can introduce either a favorable or an unfavorable outcome.
Different emphasis in overlapping contexts
Both やっと目標を達成しました。 and ついに目標を達成しました。 can report that a goal was achieved. The former can foreground overcoming waiting or difficulty; the latter can foreground the culmination.
ついに alone does not establish delight or disappointment. Read the stated result, such as 優勝 or 動かなくなる, and any further explanation of the reaction.
Is チームはついに優勝しました。 negative because it contains ついに?
Show the answer and why
No. It says that the team finally won the championship. ついに does not itself make the result unfavorable.