Question

Cho 3 số thực dương a, b, c thõa mãn a + b + c = 1. Tìm GTLN của biểu thức
\(A=\sqrt{2ab+2b} + \sqrt{2bc+2c} + \sqrt{2ca+2a}\)

Answer 1

\(A=\sqrt{2b\left(a+1\right)}+\sqrt{2c\left(b+1\right)}+\sqrt{2a\left(c+1\right)}\)

\(A=\dfrac{1}{2\sqrt{2}}.2\sqrt{4b\left(a+1\right)}+\dfrac{1}{2\sqrt{2}}.2\sqrt{4c\left(b+1\right)}+\dfrac{1}{2\sqrt{2}}.2\sqrt{4a\left(c+1\right)}\)

\(A\le\dfrac{1}{2\sqrt{2}}\left(4b+a+1\right)+\dfrac{1}{2\sqrt{2}}\left(4c+b+1\right)+\dfrac{1}{2\sqrt{2}}\left(4a+c+1\right)\)

\(A\le\dfrac{1}{2\sqrt{2}}\left[5\left(a+b+c\right)+3\right]=2\sqrt{2}\)

Dấu "=" xảy ra khi \(a=b=c=\dfrac{1}{3}\)