cho a,b,c>0 a.b.c=1 Tìm MAX M=1/(ab+a+2)+1/(bc+b+2)+1/(ca+c+2)

`1/(ab+a+2)≤1/4 (1/(ab+1)+1/(a+1))`

tương tự 

`1/(cb+b+2)≤1/4 (1/(cb+1)+1/(b+1))`

`1/(ac+c+2)≤1/4 (1/(ac+1)+1/(c+1))`

`⇒M≤1/4 (1/(ac+1)+1/(c+1))+1/4 (1/(cb+1)+1/(b+1))+1/4 (1/(ab+1)+1/(a+1))`

`⇒M≤1/4 ((ab+1)/(ab+1)+(ac+1)/(ac+1)+(bc+1)/(bc+1))`

`⇒M≤1/4 .3≤3/4`

`''=''`xẩy ra khi :

`a=b=c=1`