Áp dụng BĐT Cauchy-Schwarz ta có:
\(A=x^4+y^4+z^4=\left(x^2\right)^2+\left(y^2\right)^2+\left(z^2\right)^2\)
\(\Rightarrow\left[\left(1^2\right)^2+\left(1^2\right)^2+\left(1^2\right)^2\right]\left[\left(x^2\right)^2+\left(y^2\right)^2+\left(z^2\right)^2\right]\ge\left(x^2+y^2+z^2\right)^2\)
\(\Rightarrow3\left[\left(x^2\right)^2+\left(y^2\right)^2+\left(z^2\right)^2\right]\ge\left(x^2+y^2+z^2\right)^2\)
Lại có BĐT quen thuộc \(x^2+y^2+z^2\ge xy+yz+xz\):
\(\Rightarrow3A\ge\left(x^2+y^2+z^2\right)^2\ge\left(xy+yz+xz\right)^2=1\)
\(\Rightarrow3A\ge1\Rightarrow A\ge\dfrac{1}{3}\)
Đẳng thức xảy ra khi \(x=y=z=\dfrac{1}{\sqrt{3}}\)
