“Being your first lady has been the greatest honor of my life,” Mrs. Obama told an audience of educators in the East Room, her voice catching as her eyes shone with tears. “And I hope I’ve made you proud.”

Her 21-minute speech, an appeal for hope and inclusiveness as the salves to the forces of fear and division, carried an implicit rebuke to President-elect Donald J. Trump, whom she did not name, delivered with the quiet intensity and aspirational language that came to mark her appearances on the campaign trail last year.

“Our glorious diversity — our diversities of faiths, and colors, and creeds — that is not a threat to who we are; it makes us who we are,” Mrs. Obama said.

“So to the young people here and the young people out there: Do not ever let anybody make you feel like you don’t matter or like you don’t have a place in our American story because you do, and you have a right to be exactly who you are,” she added.