Với mọi x;y;z ta có:
\(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\)
\(\Leftrightarrow2x^2+2y^2+2z^2\ge2xy+2yz+2zx\)
\(\Leftrightarrow3x^2+3y^2+3z^2\ge x^2+y^2+z^2+2xy+2yz+2zx\)
\(\Leftrightarrow3x^2+3y^2+3z^2\ge\left(x+y+z\right)^2\)
\(\Rightarrow x+y+z\le\sqrt{3\left(x^2+y^2+z^2\right)}\)
Áp dụng:
\(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\le\sqrt{3\left(4a+1+4b+1+4c+1\right)}=\sqrt{3.7}=\sqrt{21}< 5\) (đpcm)