a) \(E=x^2-8x+15=\left(x^2-8x+16\right)-1=\left(x-4\right)^2-1\ge-1\)
\(minE=-1\Leftrightarrow x=4\)
b) \(F=-x^2+5x+6=-\left(x^2-5x+\dfrac{25}{4}\right)+\dfrac{49}{4}=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{49}{4}\le\dfrac{49}{4}\)
\(maxF=\dfrac{49}{4}\Leftrightarrow x=\dfrac{5}{2}\)
c) \(G=2x^2-6x+8=2\left(x^2-3x+\dfrac{9}{4}\right)+\dfrac{7}{2}=2\left(x-\dfrac{3}{2}\right)^2+\dfrac{7}{2}\ge\dfrac{7}{2}\)
\(minG=\dfrac{7}{2}\Leftrightarrow x=\dfrac{3}{2}\)
d) \(H=-3x^2+15x-1=-3\left(x^2-5x+\dfrac{25}{4}\right)+\dfrac{71}{4}=-3\left(x-\dfrac{5}{2}\right)^2+\dfrac{71}{4}\le\dfrac{71}{4}\)
\(minH=\dfrac{71}{4}\Leftrightarrow x=\dfrac{5}{2}\)
e) \(I=\left(2x-1\right)^2+\left(x+2\right)^2=4x^2-4x+1+x^2+4x+4=5x^2+5\ge5\)
\(minI=5\Leftrightarrow x=0\)
f) \(F=x^2+2x+7-6\left|x+1\right|=\left(x^2+2x+1\right)-6\left|x+1\right|+9-3=\left(x+1\right)^2-6\left|x+1\right|+9-3=\left(x+1-3\right)^2-3=\left(x-2\right)^2-3\ge-3\)
\(minF=-3\Leftrightarrow x=2\)
\(a,E=\left(x^2-8x+16\right)-1=\left(x-4\right)^2-1\ge-1\)
Dấu \("="\Leftrightarrow x=4\)
\(b,F=-\left(x^2-2\cdot\dfrac{5}{2}x+\dfrac{25}{4}-\dfrac{49}{4}\right)=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{49}{4}\le\dfrac{49}{4}\)
Dấu \("="\Leftrightarrow x=\dfrac{5}{2}\)
\(c,G=2\left(x^2-2\cdot\dfrac{3}{2}x+\dfrac{9}{4}-\dfrac{41}{4}\right)=2\left(x-\dfrac{3}{2}\right)^2+\dfrac{41}{2}\ge\dfrac{41}{2}\)
Dấu \("="\Leftrightarrow x=\dfrac{3}{2}\)
\(d,H=-3\left(x^2-2\cdot\dfrac{5}{2}x+\dfrac{25}{4}-\dfrac{21}{4}\right)=-3\left(x-\dfrac{5}{2}\right)^2+\dfrac{63}{4}\le\dfrac{63}{4}\)
Dấu \("="\Leftrightarrow x=\dfrac{5}{2}\)
\(e,I=\left(2x-1\right)^2+\left(x+2\right)^2\ge0\)
Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}2x-1=0\\x+2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{2}\\x=-2\end{matrix}\right.\Leftrightarrow x\in\varnothing\)
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