\(M=\dfrac{4x^2+2-4x^2+4x-1}{4x^2+2}=1-\dfrac{4x^2-4x+1}{4x^2+2}=1-\dfrac{\left(2x-1\right)^2}{4x^2+2}\)
\(M\) sẽ đạt GTLN khi \(\dfrac{\left(2x-1\right)^2}{4x^2+2}\) đạt GTNN
Do \(\left\{{}\begin{matrix}\left(2x-1\right)^2\ge0\forall x\\4x^2+2>0\forall x\end{matrix}\right.\) \(\Rightarrow\dfrac{\left(2x-1\right)^2}{4x^2+2}\ge0\) \(\forall x\)
\(\Rightarrow M\) lớn nhất khi \(\dfrac{\left(2x-1\right)^2}{4x^2+2}=0\)
\(\Rightarrow M_{max}=1-0=1\) khi \(2x-1=0\Leftrightarrow x=\dfrac{1}{2}\)