Ytukyg

begin{array}{ll} text{maximize} & prod_{i=1}^{n}x_i\ text{subject to} & mathrm sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,nend{array}

if $x_1geqcdotsgeq x_n$ and $y_1geqcdotsgeq y_nquad$ ($x_i,y_iin mathbb{R}^+$ and all $y_i$ are given).

My attempt: By induction

For $n=2$, we need to maximize $x_1x_2$ when $x_1+x_2leq y_1+y_2$ and $x_2leq y_2$. Let $x_2=y_2-tquad$ ($tgeq0$), then $x_1leq y_1+t$, thus to maximize $x_1x_2$, we let $x_1=y_1+t$. Then $x_1x_2=(y_2-t)(y_1+t)=y_1y_2-(y_1-y_2)t-t^2leq y_1y_2$. Thus $max x_1x_2=y_1y_2$ when $t=0$.

Suppose $maxlimits_{{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n}} prod_{i=1}^{n}x_i=prod_{i=1}^{n}y_i$, then begin{array}{ll}maxlimits_{{sum_{i=k}^{n+1}x_ileqsum_{i=k}^{n+1}y_i \forall k=1,2,cdots,n+1}} prod_{i=1}^{n+1}x_i=(prod_{i=1}^{n}y_i)x_{n+1}leq(prod_{i=1}^{n}y_i)y_{n+1}.end{array}

I think I cannot use induction like that because of constraints, any other method to try to prove my hypothesis that maximum is achieved when $x_i=y_i$?