Question

Tìm GTLN của A = 1 - 1/x - 1/x²

Answer 1

\(A=1-\frac{1}{x}-\frac{1}{x^2}\)

Đặt \(\frac{1}{x}=a\)

Khi đó : \(A=-a^2-a+1\)

\(A=-\left(a^2+a-1\right)\)

\(A=-\left[a^2+2\cdot a\cdot\frac{1}{2}+\frac{1}{4}-\frac{5}{4}\right]\)

\(A=-\left[\left(a+\frac{1}{2}\right)^2-\frac{5}{4}\right]\)

\(A=\frac{5}{4}-\left(a+\frac{1}{2}\right)^2\le\frac{5}{4}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow a=\frac{-1}{2}\Leftrightarrow\frac{1}{x}=\frac{-1}{2}\Leftrightarrow x=-2\)