a ) ĐKXĐ : \(x\ne-1\)
\(P=\dfrac{2x^2+3x+1}{x^3+x^2+2x+2}=\dfrac{2x\left(x+1\right)+x+1}{x^2\left(x+1\right)+2\left(x+1\right)}=\dfrac{\left(2x+1\right)\left(x+1\right)}{\left(x^2+2\right)\left(x+1\right)}=\dfrac{2x+1}{x^2+2}\)
b ) Tìm Min
\(P+\dfrac{1}{2}=\dfrac{2x+1}{x^2+2}+\dfrac{1}{2}=\dfrac{4x+2+x^2+2}{2\left(x^2+2\right)}=\dfrac{\left(x+2\right)^2}{2\left(x^2+2\right)}\)
\(\Rightarrow P=\dfrac{\left(x+2\right)^2}{x^2+2}-\dfrac{1}{2}\ge-\dfrac{1}{2}\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x=-2\)
Tìm Max
\(P-1=\dfrac{2x+1}{x^2+2}-1=\dfrac{2x+1-x^2-2}{x^2+2}=\dfrac{-\left(x-1\right)^2}{x^2+2}\)
\(\Rightarrow P=1-\dfrac{\left(x-1\right)^2}{x^2+2}\le1\forall x\)
Dấu " = " xảy ra \(\Leftrightarrow x=1\)
Vậy ...
P/s : Sử dụng Delta để làm nhé bạn :D
\(a)P=\dfrac{2x^2+3x+1}{x^3+x^2+2x+2}\\ P=\dfrac{2x^2+2x+x+1}{x^2\left(x+1\right)+2\left(x+1\right)}\\ P=\dfrac{2x\left(x+1\right)+x+1}{\left(x+1\right)\left(x^2+2\right)}\\ P=\dfrac{\left(x+1\right)\left(2x+1\right)}{\left(x+1\right)\left(x^2+2\right)}\\ P=\dfrac{2x+1}{x^2+2}\)
