Giải:
\(A=\dfrac{x^2+x+1}{\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{4\left(x^2+x+1\right)}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{4x^2+4x+4}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{3x^2+x^2+6x-2x+3+1}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{\left(3x^2+6x+3\right)+\left(x^2-2x+1\right)}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{3\left(x+1\right)^2+\left(x-1\right)^2}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{3\left(x+1\right)^2}{4\left(x-1\right)^2}+\dfrac{\left(x-1\right)^2}{4\left(x-1\right)^2}\)
\(\Leftrightarrow A=\dfrac{3\left(x+1\right)^2}{4\left(x-1\right)^2}+\dfrac{1}{4}\ge\dfrac{1}{4}\)
\(\Leftrightarrow A_{Min}=\dfrac{1}{4}\)
\("="\Leftrightarrow x=-1\)
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