Question

Cho a,b,c là ba số thực âm thỏa mãn a+b+c=1. CMR

\(\sqrt{7a+9}+\sqrt{7b+9}+\sqrt{7c+9}\ge10\)

Answer 1

số thực ko âm nhé

\(a+b+c=1\Leftrightarrow a;b;c\le1\)

\(\Leftrightarrow\left\{{}\begin{matrix}a^2\le a\\b^2\le b\\c^2\le c\end{matrix}\right.\)

\(\sqrt{7a+9}+\sqrt{7b+9}+\sqrt{7c+9}\)

\(=\sqrt{a+6a+9}+\sqrt{b+6b+9}+\sqrt{c+6c+9}\)

\(\ge\sqrt{a^2+6a+9}+\sqrt{b^2+6b+9}+\sqrt{c^2+6c+9}\)

\(=\sqrt{\left(a+3\right)^2}+\sqrt{\left(b+3\right)^2}+\sqrt{\left(c+3\right)^2}\)

\(=a+b+c+9=10\left(a;b;c\ge0\right)\)

\("="\Leftrightarrow\)a;b;c là hoán vị (0;0;1)