\(a.A=5x-x^2=-\left(x^2-5x\right)=-\left[x^2-2x.\dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2-\left(\dfrac{5}{2}\right)^2\right]=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{25}{4}\)
Vì \(\left(x-\dfrac{5}{2}\right)^2\ge0\forall x\in R\Rightarrow-\left(x-\dfrac{5}{2}\right)^2\le0\forall x\in R\)
\(\Rightarrow-\left(x-\dfrac{5}{2}\right)^2+\dfrac{25}{4}\le\dfrac{25}{4}\)
\(\Rightarrow Max_A=\dfrac{25}{4}\Leftrightarrow x=\dfrac{5}{2}\)
\(b.B=x-x^2=-\left(x^2-x\right)=-\left(x^2-2x\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2-\left(\dfrac{1}{2}\right)^2\right)=-\left[\left(x-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\right]=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\)
Vì \(\left(x-\dfrac{1}{2}\right)^2\ge0\forall x\in R\Rightarrow-\left(x-\dfrac{1}{2}\right)^2\le0\forall x\in R\)
\(\Rightarrow-\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\le\dfrac{1}{4}\)
\(\Rightarrow Max_B=\dfrac{1}{4}\Leftrightarrow x=\dfrac{1}{2}\)
\(c.C=4x-x^2+3=-\left(x^2-4x-3\right)=-\left(x^2-4x+2^2-7\right)=-\left(x-2\right)^2+7\)
Vì \(\left(x-2\right)^2\ge0\forall x\in R\Rightarrow-\left(x-2\right)^2\le0\forall x\in R\)
\(\Rightarrow-\left(x-2\right)^2+7\le7\)
\(\Rightarrow Max_B=7\Leftrightarrow x=2\)
\(d.D=-x^2+6x-11=-\left(x^2-6x+11\right)=-\left(x^2-6x+3^2+2\right)=-\left(x-3\right)^2-2\)
Vì \(\left(x-3\right)^2\ge0\forall x\in R\Rightarrow-\left(x-3\right)^2\le0\forall x\in R\)
\(\Rightarrow-\left(x-3\right)^2-2\le-2\)
\(\Rightarrow Max_D=-2\Leftrightarrow x=3\)
\(e.E=5-8x-x^2=-\left(x^2+8x-5\right)=-\left(x^2+8x+4^2-21\right)=-\left(x+4\right)^2+21\)
Vì \(\left(x+4\right)^2\ge0\forall x\in R\Rightarrow-\left(x+4\right)^2\le0\forall x\in R\)
\(\Rightarrow-\left(x+4\right)^2+21\le21\)
\(\Rightarrow Max_E=21\Leftrightarrow x=-4\)