Question

Cho \(a,b,c\ge0\) và a + b + c = 1. Tìm Max của \(F=\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c}\)

Answer 1

Áp dụng bđt Cauchy-Schwarz có:

\(F^2=\left(\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c}\right)^2\)

\(=2+2\sqrt{\left(a+b\right)\left(a+c\right)}+2\sqrt{\left(a+c\right)\left(b+c\right)}+2\sqrt{\left(a+b\right)\left(b+c\right)}\le2+2a+b+c+a+b+2c+a+c+2b\)

\(=2+4a+4b+4c=2+4\left(a+b+c\right)=6\)

\(\Rightarrow F\le\sqrt{6}\)

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

Vậy \(F_{max}=\sqrt{6}\)