2 ^x1+2^x2-2^(x1+x2)=1-1+2 ^x1+2^x2-2^x1·2^x2
=1-(1-2 ^x1)+2^x2(1-2^x1)
=1+(1-2 ^x1)(2^x2-1)
=1-(2 ^x1-1)(2^x2-1)
由条件:2 ^x1≥1,2^x2≥1(当x1=x2=0时取等号)
所以2 ^x1-1≥0,2 ^x2-1≥0
(2 ^x1-1)(2^x2-1)≥0
-(2 ^x1-1)(2^x2-1)≤0
1-(2 ^x1-1)(2^x2-1)≤1(当x1=x2=0时取等号)
所以,其最大值是1.