Question

cho các số thực dương a,b,c thỏa mãn a+b+c=1 tìm min p=\(\frac{a}{\sqrt{1-a}}+\frac{b}{\sqrt{1-b}}+\frac{c}{\sqrt{1-c}}\)

Answer 1

Ta có:\(P=\sum\frac{a}{\sqrt{1-a}}\)

\(P=\sum\frac{a}{\sqrt{b+c}}\)

\(P\ge\sum\frac{\sqrt{\frac{8}{3}}a}{b+c+\frac{2}{3}}=\sum\sqrt{\frac{8}{3}}\frac{a^2}{ab+ac+\frac{2}{3}a}\)

\(P\ge\sqrt{\frac{8}{3}}\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)+\frac{2}{3}\left(a+b+c\right)}\left(cauchy-sch\text{w}arz\right)\)

Mà \(ab+bc+ca\le\frac{\left(a+b+c\right)^2}{3}=\frac{1}{3}\)

\(\Rightarrow P\ge\sqrt{\frac{8}{3}}\frac{1}{\frac{2}{3}+\frac{2}{3}}=\sqrt{\frac{8}{3}}.\frac{3}{4}=\sqrt{\frac{3}{2}}\)

"="<=>a=b=c=1/3