Question

a,b,c>0, a+b+c=1, cmr √(a+b)+√(b+c)+√(c+a)<√6

Answer 1

Áp dụng bđt bunhiacopxki với hai bộ ba số được:

\(\left(1.\sqrt{a+b}+1.\sqrt{b+c}+1.\sqrt{c+a}\right)^2\le\left(1+1+1\right)\left[\left(\sqrt{a+b}\right)^2+\left(\sqrt{b+c}\right)^2+\left(\sqrt{c+a}\right)^2\right]\)

\(\Rightarrow\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)^2\le3\left(a+b+b+c+c+a\right)=6\)

\(\Rightarrow\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\le\sqrt{6}\)

Dấu "=" <=> a=b=c=1/3