\(N=\dfrac{4x+1}{4x^2+2}\Leftrightarrow4Nx^2-4x+2N-1=0\)
\(\Delta'=4-4N\left(2N-1\right)=-8N^2+4N+4\ge0\)
\(\Rightarrow\dfrac{-1}{2}\le N\le1\)
\(\Rightarrow N_{max}=1\) khi \(x=\dfrac{1}{2}\)
- Hoặc 1 cách làm khác:
\(N=\dfrac{4x+1}{4x^2+2}=\dfrac{4x^2+2-4x^2+4x-1}{4x^2+2}=1-\dfrac{\left(2x-1\right)^2}{4x^2+2}\)
\(N_{max}\) khi \(\dfrac{\left(2x-1\right)^2}{4x^2+2}\) đạt min
Mà \(\dfrac{\left(2x-1\right)^2}{4x^2+2}\ge0\) \(\Rightarrow N_{max}=1-0=1\) khi \(\dfrac{\left(2x-1\right)^2}{4x^2+2}=0\Leftrightarrow x=\dfrac{1}{2}\)