If this song isn't obscure, the picture in which it premiered more than likely is. "Keep Young and Beautiful" was introduced by no less a personage than Eddie Cantor in the picture Roman Scandals, one of a half-dozen movies Eddie made at Samuel Goldwyn Studios. It's certainly one of the most energetic songs on this list, with a upbeat rhythm and an obscenely memorable melody to go along with it -- I've had this song stuck in my head for weeks at a time.
Yet although it's among the best songs in Harry Warren's oeuvre, it isn't nearly as well known as many of his other tunes -- aside from an Annie Lennox cover in the early 90s, it has largely faded into obscurity. Much of this is due, I suspect, to lyrics that could easily be construed as sexist -- e.g. "What's cute about a cutie? / It's her beauty, not brains!" -- and an accompanying production number that could even more easily be construed as racist (blackface, for good reasons, isn't held in very high regard these days). Even so, you oughtn't let trifling matters like political correctness keep you from enjoying a damned good song.