设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证 x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)≥ 1/(n+1)
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![设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证 x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)≥ 1/(n+1)](/uploads/image/z/11296451-11-1.jpg?t=%E8%AE%BEx1+x2+%E2%80%A6%E2%80%A6xn%E5%B1%9E%E4%BA%8ER%2B+%E4%B8%94x1%2Bx2%2B%E2%80%A6%E2%80%A6%2Bxn%3D1%E6%B1%82%E8%AF%81+x1%5E2%2F%281%2Bx1%29+%2Bx2%5E2%2F%281%2Bx2%29%2B%E2%80%A6%E2%80%A6%2Bxn%5E2%2F%281%2Bxn%29%E2%89%A5+1%2F%28n%2B1%29)
设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证 x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)≥ 1/(n+1)
设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证
x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)≥
1/(n+1)
设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证 x1^2/(1+x1) +x2^2/(1+x2)+……+xn^2/(1+xn)≥ 1/(n+1)
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)
[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