a＞0,b＞0,a+b=1 求证（a+1/a）*（b+1/b）≥25/4

a＞0,b＞0,a+b=1 求证（a+1/a）*（b+1/b）≥25/4
a＞0,b＞0,a+b=1 求证（a+1/a）*（b+1/b）≥25/4

a＞0,b＞0,a+b=1 求证（a+1/a）*（b+1/b）≥25/4
[a+(1/a)]*[b+(1/b)]=ab+(a/b)+(b/a)+(1/ab)
≥ab+2√[(a/b)(b/a)]+[(a+b)/ab]
=ab+2+(1/b)+(1/a)
=ab+2+[(a+b)/b]+[(a+b)/a]
=ab+2+[(a/b)+1]+[1+(b/a)]
=ab+4+(a/b)+(b/a)
≥ab+4+2√[(a/b)(b/a)]
=ab+6
因为1=a+b≥2√(ab)
使用,ab≤(1/2)²=1/4
所以,原式≥(1/4)+6=25/4