Đặt \(\left(x;y;z\right)=\left(a-4;b-5;c-6\right)\) \(\Rightarrow x;y;z\ge0\)
\(\left(x+4\right)^2+\left(y+5\right)^2+\left(z+6\right)^2=90\)
\(\Leftrightarrow x^2+y^2+z^2+8x+10y+12z=13\)
\(\Leftrightarrow x^2+y^2+z^2+2xy+2xz+2yz+12\left(x+y+z\right)=13+2\left(xy+xz+yz\right)+4x+2y\)
\(\Leftrightarrow\left(x+y+z\right)^2+12\left(x+y+z\right)=13+2\left(xy+xz+yz\right)+2\left(2x+y\right)\ge13\)
\(\Leftrightarrow\left(x+y+z\right)^2+12\left(x+y+z\right)-13\ge0\)
\(\Leftrightarrow\left(x+y+z+13\right)\left(x+y+z-1\right)\ge0\)
\(\Leftrightarrow x+y+z\ge1\)
\(\Leftrightarrow a-4+b-5+c-6\ge1\)
\(\Leftrightarrow a+b+c\ge16\)
\(\Rightarrow P_{min}=16\) khi \(\left(x;y;z\right)=\left(0;0;1\right)\) hay \(\left(a;b;c\right)=\left(4;5;7\right)\)
