Question

Tìm GTNN : \(y=\dfrac{x^2+2x-1}{2x^2+4x+9}\)

Answer 1

y đạt GTNN \(\Leftrightarrow\) \(\dfrac{1}{y}\) đạt GTLN

Ta có: \(\dfrac{1}{y}=\dfrac{2x^2+4x+9}{x^2+2x-1}\)

\(\dfrac{1}{y}=\dfrac{2\left(x^2+2x-1\right)+11}{x^2+2x+1}\)

\(\dfrac{1}{y}=\dfrac{2\left(x^2+2x-1\right)}{x^2+2x-1}+\dfrac{11}{x^2+2x-1}\)

\(\dfrac{1}{y}=2+\dfrac{11}{\left(x+1\right)^2-2}\) \(\ge\) -3,5

Dấu " =" xảy ra\(\Leftrightarrow\) (x+1)² =0 \(\Leftrightarrow\) x=-1

Vậy GTNN của y là \(\dfrac{-1}{3,5}=\dfrac{-2}{7}\)