已知a＞0，b＞0，且a+b=1，求证：(1+1a)(1+1b)≥9．

Question

已知a＞0，b＞0，且a+b=1，求证：(1+(1)/(a))(1+(1)/(b))≥9．

Answer 1

∵a＞0，b＞0，且a+b=1
∴(1+(1)/(a))(1+(1)/(b))=(1+(a+b)/(a))(1+(a+b)/(b))
=(2+(b)/(a))(2+(a)/(b))=4+(2a)/(b)+(2b)/(a)+(b)/(a)×(a)/(b)
=5+(2b)/(a)+(2a)/(b)≥5+2√((2b)/(a)×(2a)/(b))＝5+4＝9
当且仅当(2b)/(a)＝(2a)/(b)，即a=b=(1)/(2)时取“=”号．
故原题得证．