cho a,b,c>0 a.b.c=1 Tìm MAX M=1/(ab+a+2)+1/(bc+b+2)+1/(ca+c+2) 15/07/2021 Bởi Lyla 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` Bình luận
`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`