\(A=x^2-20x+101\)
\(A=x^2-2\cdot x\cdot10+100+1\)
\(A=\left(x-10\right)^2+1\ge1\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=10\)
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\(B=4a^2+4a+2\)
\(B=4a^2+4a+1+1\)
\(B=\left(2a+1\right)^2+1\ge1\forall a\)
Dấu "=" xảy ra \(\Leftrightarrow a=\frac{-1}{2}\)
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\(C=x^2-4xy+5y^2+10x-22y+28\)
\(C=x^2-4xy+4y^2+y^2+10x-22y+28\)
\(C=\left(x-2y\right)^2+2\cdot\left(x-2y\right)\cdot5+25+y^2-2y+1+2\)
\(C=\left(x-2y+5\right)^2+\left(y-1\right)^2+2\ge2\forall x;y\)
Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x-2y+5=0\\y-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)
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\(D=4x-x^2+3\)
\(D=-\left(x^2-4x-3\right)\)
\(D=-\left(x^2-4x+4-7\right)\)
\(D=-\left[\left(x-2\right)^2-7\right]\)
\(D=7-\left(x-2\right)^2\le7\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=2\)
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\(E=x-x^2\)
\(E=-\left(x^2-x\right)\)
\(E=-\left(x^2-2\cdot x\cdot\frac{1}{2}+\frac{1}{4}-\frac{1}{4}\right)\)
\(E=-\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right]\)
\(E=\frac{1}{4}-\left(x-\frac{1}{2}\right)^2\le\frac{1}{4}\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=\frac{1}{2}\)
a, \(A=x^2-20x+101=x^2-2.x.10+10^2+1\)
\(=\left(x-10\right)^2+1\ge1\)
Dấu "=" xảy ra \(\Leftrightarrow\left(x-10\right)^2=0\)
\(\Leftrightarrow x-10=0\)
\(\Leftrightarrow x=10\)
Vậy : \(A_{min}=1\Leftrightarrow x=10\)
b) \(B=4a^2+4a+2=\left(2a\right)^2+2.2a.1+1^2+1\)
\(=\left(2a+1\right)^2+1\ge1\)
Dấu "=" xảy ra \(\Leftrightarrow\left(2a+1\right)^2=0\)
\(\Leftrightarrow2a+1=0\)
\(\Leftrightarrow2a=-1\)
\(\Leftrightarrow a=-\frac{1}{2}\)
Vậy : \(B_{min}=1\Leftrightarrow x=-\frac{1}{2}\)
