a) \(A=4x-x^2+3\)
\(\Leftrightarrow A=-x^2+4x-4+7\)
\(\Leftrightarrow A=-\left(x^2-4x+4\right)+7\)
\(\Leftrightarrow A=-\left(x-2\right)^2+7\)
Vậy GTLN của \(A=7\) khi \(x-2=0\Leftrightarrow x=2\)
b) \(B=2x^2-6x\)
\(\Leftrightarrow B=2x^2-6x+\dfrac{9}{2}-\dfrac{9}{2}\)
\(\Leftrightarrow B=2\left(x^2-3x+\dfrac{9}{4}\right)-\dfrac{9}{2}\)
\(\Leftrightarrow B=2\left[x^2-2.x.\dfrac{3}{2}+\left(\dfrac{3}{2}\right)^2\right]-\dfrac{9}{2}\)
\(\Leftrightarrow B=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\)
Vậy GTNN của biểu thức \(B=\dfrac{-9}{2}\) khi \(x-\dfrac{3}{2}=0\Leftrightarrow x=\dfrac{3}{2}\)
c) \(D=2x-2x^2-5\)
\(\Leftrightarrow D=-2x^2+2x-\dfrac{1}{2}-\dfrac{9}{2}\)
\(\Leftrightarrow D=-\left(2x^2-2x+\dfrac{1}{2}\right)-\dfrac{9}{2}\)
\(\Leftrightarrow D=-2\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{9}{2}\)
\(\Leftrightarrow D=-2\left[x^2-2.x.\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2\right]-\dfrac{9}{2}\)
\(\Leftrightarrow D=-2\left(x-\dfrac{1}{2}\right)^2-\dfrac{9}{2}\)
Vậy GTLN của \(D=\dfrac{-9}{2}\) khi \(x-\dfrac{1}{2}=0\Leftrightarrow x=\dfrac{1}{2}\)
