Question

证明X+1/X在（1,正无穷）中是增函数

Answer 1

设x1＜x2,且x1,x2∈(1,+∞),

f(x1)-f(x2)=x1+1/x1-x2-1/x2=(x1-x2)+(x2-x1)/x1x2=(x1-x2)(1-1/x1x2)=(x1-x2)(x1x2-1)/x1x2,

∵x1＜x2,且x1,x2∈(1,+∞),

∴x1-x2＜0,x1x2-1＞0,x1x2＞1＞0,

则f(x1)-f(x2)＜0,

即f(x1)＜f(x2)

∴f(x)=x+1/x在∈(1,+∞)上是增函数