已知lg3x+lgy=lg(x+y+1) ①求xy的最小值；②求x+y的最小值.

题目

已知lg3x+lgy=lg(x+y+1) ①求xy的最小值；②求x+y的最小值.

答案

lg3x+lgy=lg(x+y+1) =>
lg(3x*y) = lg(x+y+1)
且3x>0, y >0 x+y+1 > 0
3xy = x + y + 1
x(3y+1) = 1
x = 1/(3y+1)
xy = y/(3y + 1) = (y + 1/3 - 1/3)/(3y+1) =
1/3 - 1/(9y+3) > 1/3
xy无最小值,极限为1/3
x+y = 3xy -1 > 0
x+y无最小值,极限为0