Question

证明函数y=x+1/x在区间[1,+∞]上是增函数.

Answer 1

证明：任取x1>x2>1

(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：上式>0

即：x1+1/x1>x2+1/x2

所以函数在[1,+∞）上是增函数.