设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证 x1^2/(1+x1) +x2^2

网友回答

1.[n+1][1/(1+x1) +1/(1+x2)+……+1/(1+xn)]=
=[(1+x1) +(1+x2)+……+(1+xn)][1/(1+x1) +1/(1+x2)+……+1/(1+xn)]≥n^2
==> 1/(1+x1) +1/(1+x2)+……+1/(1+xn)≥n^2/(n+1)
2.x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)=
=[x1-1+1/(1+x1)]+[x2-1+1/(1+x2)]+...[xn-1+1/(1+xn)]=
=1-n+1/(1+x1) +1/(1+x2)+……+1/(1+xn)≥
≥1-n+n^2/(n+1)=1/(n+1)
3.x1=x2=……=xn=1/n
==> x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)=1/(n+1)
======以下答案可供参考======
供参考答案1:
[x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)](n+1)=[x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)][(1+x1) +(1+x2)+……+(1+xn)]≥ 1