\(E=x^2+x+1\)
\(=x^2+2.\frac{1}{2}x+\frac{1}{4}+\frac{3}{4}\)
\(=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\forall x\) (do \(\left(x+\frac{1}{2}\right)^2\ge0\forall x\)
\(\Rightarrow Min_E=\frac{3}{4}\Leftrightarrow x=-\frac{1}{2}\)
\(F=2x^2-4x+5\)
\(=2\left(x^2-2x+1\right)+3\)
\(=2\left(x-1\right)^2+3\ge3\forall x\) (do...)
\(\Rightarrow Min_F=3\Leftrightarrow x-1=0\Leftrightarrow x=1\)
\(B=1-x^2+3x\)
\(=-\left(x^2-2.\frac{3}{2}x+\frac{9}{4}\right)+\frac{9}{4}+1\)
\(=-\left(x-\frac{3}{2}\right)^2+\frac{13}{4}\le\frac{13}{4}\forall x\) (do...)
\(\Rightarrow Max_F=\frac{13}{4}\Leftrightarrow x=\frac{3}{2}\)
