They the incident to find the guilty

If your options are investigate and investigated, I'd definitely go for the past tense - investigated.
It means that the action happened in the past, before the moment this sentence is spoken/written. If you choose the present tense (They investigate the incident...), it sounds wrong because they don't do it every day (which is something that tense would suggest). Instead of investigate, you could use are investigating, but since that isn't a choice, you should choose investigated.