Question

[vung nguyen thi]

Cho a,b,c dương thỏa mãn a+b+c=1

Tìm GTLN của P=\(\dfrac{ab}{\sqrt{c+ab}}+\dfrac{bc}{\sqrt{a+bc}}+\dfrac{ca}{\sqrt{b+ca}}\)

Answer 1

[Unruly Kid]

\(\sum\dfrac{ab}{\sqrt{c+ab}}=\sum\dfrac{ab}{\sqrt{c\left(a+b+c\right)+ab}}=\sum\dfrac{ab}{\sqrt{\left(c+a\right)\left(c+b\right)}}\le\dfrac{1}{2}\sum\left(\dfrac{ab}{a+b}+\dfrac{ab}{a+c}\right)=\dfrac{a+b+c}{2}=\dfrac{1}{2}\)

GTNN của P là \(\dfrac{1}{2}\Leftrightarrow a=b=c=\dfrac{1}{3}\)