The question is not about the two adjectives, but the repetition of the determiner "most" before each adjective. There isn't a right or wrong here, but context could have a bearing on which you used.
As a general rule, the most succinct way of phrasing something is the best. If you don't need to repeat a word, then don't - it makes sentences longer than they need to be and makes reading a chore.
Having said that, there are situations where you might need to repeat a determiner as in your example. For example, let's say a movie wins two awards at the Oscars - an award for 'Best Story', and an award for 'Best Cinematography'. Because these are names for the awards, you would say "the film won Best Story and Best Cinematography".
In your example, "most active" and "most energetic" might not be names - but if there was actually a formal measurement of these things, it might be appropriate to repeat the determiner "most", otherwise it might just sound like you are expressing an opinion.
Also remember that if two metrics are closely linked then you likely could measure them together - certainly "active and energetic" seem like they could be observed from one activity and probably trend in the same direction (ie being the most energetic also makes one the most active), so it makes sense to say "the most active and energetic". It wouldn't be wrong to say "the most active and popular", for example, but as these things could be unrelated I think I would probably say "the most active and the most popular".