Question

Cho a, b, c > 0 thỏa mãn : a+b+c = 1 .

Tìm Max P= \(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\)

Answer 1

Áp dụng BĐT AM - GM, ta có \(\sqrt{\dfrac{2}{3}}P=\sqrt{\dfrac{2}{3}\left(a+b\right)}+\sqrt{\dfrac{2}{3}\left(b+c\right)}+\sqrt{\dfrac{2}{3}\left(c+a\right)}\le\dfrac{\dfrac{2}{3}+a+b}{2}+\dfrac{\dfrac{2}{3}+b+c}{2}+\dfrac{\dfrac{2}{3}+c+a}{2}=\dfrac{2+2\left(a+b+c\right)}{2}=2\Rightarrow P\le\sqrt{6}\)Dấu = xảy ra <=> \(a=b=c=\dfrac{1}{3}\)