Question

Cho a,b,c>0 thỏa mãn a+b+c=1 .CMR:

\(\sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}\) \(\le\)\(\sqrt{6}\)

(Sử dụng Cauchy)

Answer 1

Lời giải:

Áp dụng BĐT Cauchy cho các số dương ta có:

\(\sqrt{\frac{2}{3}(1-a)}\leq \frac{\frac{2}{3}+1-a}{2}\)

\(\sqrt{\frac{2}{3}(1-b)}\leq \frac{\frac{2}{3}+1-b}{2}\)

\(\sqrt{\frac{2}{3}(1-c)}\leq \frac{\frac{2}{3}+1-c}{2}\)

Cộng theo vế:

\(\sqrt{\frac{2}{3}}(\sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c})\leq \frac{2+3-(a+b+c)}{2}\)

\(\Leftrightarrow \sqrt{\frac{2}{3}}(\sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c})\leq 2\)

\(\Leftrightarrow \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}\leq \sqrt{6}\)

Ta có đpcm

Dấu bằng xảy ra khi \(a=b=c=\frac{1}{3}\)