22x?xx121?????2lnx?x?2x?2?2lnx?x?2x?2??2ln?x?x?2lnx?2lnx ????2???112?x1x2x2212122222221?x2x1xx1??2ln1.令t?1?0?t?1?，h?t???t?2lnt，则x1x2x2x2t212?t2?2t?1??t?1?h'?t???2?1????0，即h?t?在?0,1?上单调递减. 22tttt因为a?2e?2a1，所以x1?x2??e?，

2ee2x12?x2?2x1x211???e??2， ??e??，即x1x2ee??2?x?x2?所以1x1x2所以

2x1x2111??e?，即t??e?，

x2x1ete?1?t?e??t???0，0?t?1， 所以??e?所以0?t?1. e因为h?t?在?0,?上单调递减，

e??1??所以h?t?的最小值为h???e?即f?x1??f?x2?的最小值为e?

?1??e?1?2， e1?2. e