1)
What is the meaning of "Ba'avur ha'Yeled ha'Chai"?
1.
Radak citing Targum Yonasan: While the child was alive.
2.
Radak: For the sake of the live child.
3.
Malbim: Since you were pained only as long as the baby lived, it seems that you were pained that he lives, for you wanted him to die 1 !
1
Malbim (11:3): According to Mahari, the child was a Mamzer. David would not pray and fast that he live! Refer to 11:3:2:3**.
2)
Did his servants not understand that fasting can help only as long as the child is alive?
1.
Malbim: They thought that the fasting was merely due to pain; Tefilah sufficed by itself. They did not realize that the [merit] of the fast could help his Tefilah.