Question

若a＞0,b＞0,a＋b＝1，求证(1) (1＋1／a)(1＋1／b)≥9 (2)a^4＋b^4≥1／8

详细的问题说明，有助于回答者给出准确的答案

Answer 1

(1+1/a)(1+1/b)
=(a+1)/a*(b+1)/b
=(ab+a+b+1)/ab
=(ab+2)/ab
=1+2/ab

(a-b)^2>=0
(a+b)^2>=4ab
ab<=[(a+b)/2]^2=1/4
2/ab>=8

(1+1/a)(1+1/b)>=1/8
a=b=1/2时，取等号。仅供参考

Answer 2

(1+1/a)(1+1/b)
=(a+1)/a*(b+1)/b
=(ab+a+b+1)/ab
=(ab+2)/ab
=1+2/ab

(a-b)^2>=0
(a+b)^2>=4ab
ab<=[(a+b)/2]^2=1/4
2/ab>=8

(1+1/a)(1+1/b)>=1/8
a=b=1/2时，取等号。