Áp dụng BĐT Cauchy-Schwarz ta có:
\(\left(3+1\right)\left(3x^2+y^2\right)\ge\left(3x+y\right)^2\)
\(\Rightarrow4\left(3x^2+y^2\right)\ge\left(3x+y\right)^2\)
\(\Rightarrow4\left(3x^2+y^2\right)\ge\left(3x+y\right)^2=1^2=1\)
\(\Rightarrow M=3x^2+y^2\ge\dfrac{1}{4}\)
Đẳng thức xảy ra khi \(x=y=\dfrac{1}{4}\)