Question

Cho x, y, z dương thoả mãn \(x\ge2019;y\ge2020;z\ge2021\)

tìm GTNN của \(M=x+\frac{1}{x}-\frac{1}{2019}+y+\frac{1}{y}-\frac{1}{2020}+z+\frac{1}{z}-\frac{1}{2021}\)

Answer 1

Ta có:

\(x+\frac{1}{x}=\left(x+\frac{2019^2}{x}\right)-\frac{2019^2-1}{x}\ge_{Cauchy}2\sqrt{x.\frac{2019^2}{x}}-\frac{2019^2-1}{2019}=2.2019-2019+\frac{1}{2019}=2019+\frac{1}{2019}\).

Tương tự, \(y+\frac{1}{y}\ge2020+\frac{1}{2020};z+\frac{1}{z}\ge2021+\frac{1}{2021}\).

Do đó: \(M\ge2019+2020+2021=3.2020=6060\).

Dấu "="xảy ra khi và chỉ khi \(\left\{{}\begin{matrix}x=2019\\y=2020\\z=2021\end{matrix}\right.\)