Đặt \(M\left(x;y;z\right)\Rightarrow\overrightarrow{AM}=\left(x-7;y-2;z-3\right)\)
\(\Rightarrow AM=\sqrt{\left(x-7\right)^2+\left(y-2\right)^2+\left(z-3\right)^2}\)
Tương tự:
\(BM=\sqrt{\left(x-1\right)^2+\left(y-4\right)^2+\left(z-3\right)^2}\)
\(CM=\sqrt{\left(x-1\right)^2+\left(y-2\right)^2+\left(z-6\right)^2}\)
\(\sqrt{3}MD=\sqrt{3}.\sqrt{\left(x-1\right)^2+\left(y-2\right)^2+\left(z-3\right)^2}\)
\(P\ge\sqrt{\left(x-7\right)^2}+\sqrt{\left(y-4\right)^2}+\sqrt{\left(z-6\right)^2}+\sqrt{\left(x+y+z-6\right)^2}\)
\(=\left|7-x\right|+\left|4-y\right|+\left|6-z\right|+\left|x+y+z-6\right|\)
\(\ge\left|17-6\right|=11\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}x-1=0\\y-2=0\\z-3=0\end{matrix}\right.\) \(\Rightarrow M\left(1;2;3\right)\)
