构造柯西不等式,用取等号条件解题

摘要

我们知道,柯西不等式:a_i,b_i∈R,则sum from i=1 to n a_i2 sum from i=1 to n b_i2≥（sum from i=1 to n a_ib_i）2……（1）当且仅当a_i=kb_i（i=1,2,…,n）不等式等号成立。它可以作如下变形: 由（1）得（sum from i=1 to n a_i2 sum from i=1 to n b_i2）1/2≥sum from i=1 to n a_ib_i,添项变为sum from i=1 to n a_i2+2 （sum from i=1 to n a_i2 sum from i=1 to n b_i2）1/2+sum from i=1 to n b_i2≥sum from i=1 to n a_i2+2sum from i=1 to n a_ib_i+sum from i=1 to n b_i2,或sum from i=1 to n a_i2-2 （sum from i=1 to n a_i2 sum from i=1 to n b_i2）1/2+sum from i=1 to n b_i2≤sum from i=1 to n a_i2-2 sum from i=1 to n a_i b_i+sum from i=1 to n b_i2,分别配方,并开方转