a. ĐKXĐ: \(x\ne1\)
\(C=\left(\frac{x^2+2}{x^3-1}+\frac{x+1}{x^2+x+1}-\frac{1}{x-1}\right):\frac{x^2+1}{x^2+x+1}\)
\(=\left(\frac{x^2+2+\left(x+1\right)\left(x-1\right)-\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\right).\frac{x^2+x+1}{x^2+1}\)
\(=\frac{x^2+2+x^2-1-x^2-x-1}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+x+1}{x^2+1}\)
\(=\frac{x^2-x}{\left(x-1\right)\left(x^2+x+1\right)}.\frac{x^2+x+1}{x^2+1}\)
\(=\frac{x}{x^2+1}\)
b. \(\left|x+1\right|=2\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1=2\\x+1=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}C=\frac{1}{1+1}=\frac{1}{2}\\C=\frac{-3}{\left(-3\right)^2+1}=\frac{-3}{10}\end{matrix}\right.\)
c. Bài này mình chỉ tìm dược max thôi, không tìm thấy min.
\(C=\frac{x}{x^2+1}=\frac{2x}{2\left(x^2+1\right)}=\frac{x^2+1-x^2+2x-1}{2\left(x^2+1\right)}=\frac{1}{2}-\frac{\left(x-1\right)^2}{2\left(x^2+1\right)}\le\frac{1}{2}\)
\(\Rightarrow C_{max}=\frac{1}{2}\) khi \(x=1\)
\(C=\frac{2x}{2\left(x^2+1\right)}=\frac{-x^2-1+x^2+2x+1}{2\left(x^2+1\right)}=-\frac{1}{2}+\frac{\left(x+1\right)^2}{2\left(x^2+1\right)}\ge-\frac{1}{2}\)
\(\Rightarrow C_{min}=-\frac{1}{2}\) khi \(x=-1\)
