证明：设x1>x2>1
f(x1)-f(x2)
=(x1+1/x1)-(x2+1/x2)
=(x1^2+1)/x1-(x2^2+1)/x2=(x1^2x2+x2-x1x2^2-x1)/(x1x2)
=[x1(x1x2-1)-x2(x1x2-1)]/(x1x2)=[(x1x2-1)(x1-x2)]/(x1x2)
因为：x1>x2>1,所以：x1x2>1,x1-x2>0
所以：f(x1)>f(x2)
所以函数在[1,+∞）上是增函数.