a.b. \(A=\dfrac{2}{x-1}+\dfrac{2\left(x+1\right)}{x^2+x+1}+\dfrac{x^2-10x+3}{x^3-1}\) ( x ≠ 1 )
\(A=\dfrac{2\left(x^2+x+1\right)+2\left(x+1\right)\left(x-1\right)+x^2-10x+3}{x^3-1}\)
\(A=\dfrac{2x^2+2x+2+2x^2-2+x^2-10x+3}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(A=\dfrac{5x^2-8x+3}{\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{5x^2-5x-3x+3}{\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{5x\left(x-1\right)-3\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{\left(x-1\right)\left(5x-3\right)}{x^2+x+1}=\dfrac{5x-3}{x^2+x+1}\)
c.
\(A=\dfrac{5x-3}{x^2+x+1}\)
\(\Leftrightarrow A\left(x^2+x+1\right)=5x-3\)
\(\Leftrightarrow Ax^2+Ax+A-5x+3=0\)
\(\Leftrightarrow Ax^2+\left(A-5\right)x+A+3=0\)
( \(a=A,b=A-5,c=A+3\) )
* A = 0 \(\Rightarrow x=\dfrac{3}{5}\)
* \(A\ge0\)
\(\Rightarrow\Delta=b^2-4ac\ge0\)
\(\Rightarrow\left(A-5\right)^2-4.A\left(A-3\right)\ge0\)
\(\Rightarrow A^2-10A+25-4A^2-12A\ge0\)
\(\Rightarrow-3A^2-22A+25\ge0\)
\(\Rightarrow-\dfrac{25}{4}\le A\le1\)
\(\Rightarrow Min_A=-\dfrac{25}{3}\Leftrightarrow x=\dfrac{-b}{2a}=\dfrac{\dfrac{25}{3}+5}{2.\left(\dfrac{-25}{3}\right)}=-\dfrac{4}{5}\)
