It is also your responsibility to be sure they are able to carry out their duties or need supervision and training to be able to do so
My attitude as a manager was always "praise in public, correct in private". That and it's my job to stop the crap flowing downhill to my team - unless they really, really need to know what is about to deluge them..
(Add in responsibility to mentor - no-one starts out with the skills, experience and attitude needed to do the job and part of the managers job it to teach and encourage their staff - even if it means those staff end up with the skills and confidence to leave for a better job. I don't subscribe to the "the good of the company always comes first philosophy" unless it's in the sense that encoraging people to grow into their jobs helps both performance and morale.)