I found a Joseph Carpenter, born in 1793 in Virginia living in Seneca, Monroe County, Ohio in the 1850 census. The 1850 census doesn't give relationships, but it lists with him:
Elizabeth  50  Pennsylvania
Louis         21
Elizabeth   19
Mary           17
Sarah         17
Alexander   11
Except for Elizabeth, all born in Ohio. There is also a John D. Carpenter, 30, in the next household, with:
Louisa   25
George   5
Nancy     4
Vincent   2
Elias       1
Again, all born in Ohio. Next household:
David   30
Catherine  19
Mary E   7/12
All in Ohio. Next household:
Vincent   34
Eleanor   20
Randolph 11
John        10
Elijah        8
Abel          2
All born in Ohio. Next household:
George     33
Jane          27
John          8
Joseph       6
William       1
Maybe Alexander is the youngest you are seeking. I will check the 1860 or 1870 census.
Kathleen