求函数y=(x^4+x^2+5)/(x^2+1)^2的最大值与最小值

题目

求函数y=(x^4+x^2+5)/(x^2+1)^2的最大值与最小值

答案

y=(x^4+x^2+5)/(x^4+2x^2+1)
=(x^4+2x^2+1-x^2+4)/(x^4+2x^2+1)
=1+(4-x^2)/(x^4+2x^2+1)
=1+(5-(x^2+1))/(x^2+1)^2
=1-1/(x^2+1)+5/(x^2+1)^2
令t=1/(x^2+1) ,0