Question

Cho a;b;c > 0 thỏa mãn a + b + c = 1.

Tìm GTLN của biểu thức: \(T=\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}\)

Answer 1

Áp dụng BĐT Cauchy cho 2 số dương:

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

CMTT: \(\sqrt{2b+c}\le\dfrac{2b+c+1}{2},\sqrt{2c+a}\le\dfrac{2c+a+1}{2}\)

\(\Rightarrow T=\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}\le\dfrac{2a+b+1+2b+c+1+2c+a+1}{2}=\dfrac{3\left(a+b+c\right)+3}{2}=\dfrac{3+3}{2}=\dfrac{6}{2}=3\)

\(maxT=3\Leftrightarrow2a+b=2b+c=2c+a=1=a+b+c\)

\(\Leftrightarrow a=b=c=\dfrac{1}{3}\)