Question

设x1 x2 ……xn属于R+ 且x1+x2+……+xn=1求证

Answer 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)