Question

\(\frac{X^2-2x+2020}{X^2}\) TÌM GIÁ TRỊ NHỎ NHẤT NHÉ

Answer 1

\(P=\frac{x^2-2x+2020}{x^2}\)

\(P=1-\frac{2}{x}+\frac{2020}{x^2}\)

Đặt \(\frac{1}{x}=a\)

\(P=1-2a+2020a^2\)

\(P=2020\left(a^2-\frac{1}{1010}a+\frac{1}{2020}\right)\)

\(P=2020\left(a^2-2\cdot a\cdot\frac{1}{2020}+\frac{1}{2020^2}+\frac{1}{2020}-\frac{1}{2020^2}\right)\)

\(P=2020\left[\left(a-\frac{1}{2020}\right)^2+\frac{2019}{2020^2}\right]\)

\(P=2020\left(a-\frac{1}{2020}\right)^2+\frac{2019}{2020}\ge\frac{2019}{2020}\forall a\)

Dấu "=" xảy ra \(\Leftrightarrow a=\frac{1}{2020}\Leftrightarrow x=2020\)