YLL討論網

x+y+z = 1/x+1/y+1/z = xy+xz+yz
問: 當 x+y=α , xy+2z=β ,有最大值 (α , β)=?

x+y+z
= 1/x+1/y+1/z =(xy+xz+yz )/xyz
= xy+xz+yz

xyz=1

let m=xy=1/z
let x+y=n
z=1/m

n+1/m
= m+n/m

mn+1=mm+n
mm-nm +n-1=0
m=(n-1) or 1

(1) if  m=xy=1
xy+2z=β=1+2=3
x+y 隨便(>=2 or <=-2)

(2) if m=n-1
xy=x+y-1
(x-1)(y-1)=0

xy+2z=y+2/y

to maximize m+2/m,
if m>sqrt(2), we should maximize m
that is, maximize n-1

==> n=infinite, m=infinite, α =infinite, β=infinite

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