Question

Các số thực x, y, z thỏa mãn x² + y² + z² - 2 x + 4 y - 6 z = 15 Chứng minh rằng: |2 x - 3 y + 4 z - 20| ≤ 29

Answer 1

\(x^2+y^2+z^2-2x+4y-6z=15\)

\(\Leftrightarrow\left(x-1\right)^2+\left(y+2\right)^2+\left(z-3\right)^2=29\)

Đặt \(P=\left|2x-3y+4z-20\right|=\left|2\left(x-1\right)-3\left(y+2\right)+4\left(z-3\right)\right|\)

\(P^2=\left[2\left(x-1\right)-3\left(y+2\right)+4\left(z-3\right)\right]^2\)

\(P^2\le\left(2^2+3^2+4^2\right)\left[\left(x-1\right)^2+\left(y+2\right)^2+\left(z-3\right)^2\right]=29^2\)

\(\Rightarrow P\le29\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}\left(x-1\right)^2+\left(y+2\right)^2+\left(z-3\right)^2=29\\\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{4}\end{matrix}\right.\)