令x1>x2>=1
f(x1)-f(x2)=x1²+2/x1-x2²-2/x2
=(x1³x2+2x2-x1x2³-2x1)/(x1x2)
顯然分母x1x2>0
分子=x1³x2+2x2-x1x2³-2x1
=x1x2(x1²-x2²)-2(x1-x2)
=x1x2(x1+x2)(x1-x2)-2(x1-x2)
=(x1-x2)[x1x2(x1+x2)-2]
x1>x2,所以x1-x2>0
x1>1,x2>=1,所以x1x2>1
且x1+x2>1+1=2
所以x1x2(x1+x2)>2
x1x2(x1+x2)-2>0
所以分子大于0
所以f(x1)-f(x2)>0
即x1>x2>=1時(shí),f(x1)>f(x2)
所以是增函數(shù)