The words 頑張る and 最善を尽くす both means "do my best". What's the difference and when should I use each of them?
頑張る sounds more "making strenuous effort" which can be positive or negative. It can imply you are giving it your best shot or you are just wasting energy/trifling in a different context.
However, 最善を尽くす sounds one is trying to find the best scenario as much as possible. It does not sound like "brute-force".
Probably, in a medical situation, saying 頑張る : "do one's best" does not take patients to be comfortable to receive the service by doctors. I think 最善を尽くす : "using the best solutions as much as they can" relieving patients more.
頑張る does not have the meaning of "my best". ALC gives 20 possible translations of 頑張る, which do not include "do my best". It is just "to work hard".
頑張る is fairly common, and 最善を尽くす is a big and serious set phrase you should use sparingly.