Question

[Nguyễn Thảo Nguyên]

CMR: 

      A = x² + y² - 2(x + y) + 5  > 0 với mọi x, y

Answer 1

[Edogawa Conan]

Ta có: A = x² + y²  - 2(x + y) + 5

A = x² + y² - 2x - 2y + 5

A = (x² - 2x +1) + (y² - 2y + 1) + 3

A = (x - 1)² +  (y - 1)² + 3

Do (x - 1)² \(\ge\)0 \(\forall\)x; (y - 1)² \(\ge\)0 \(\forall\)y

=> (x - 1)² + (y - 1)² + 3 \(\ge\)3 > 0 \(\forall\)x;y

=> A > 0  \(\forall\)x; y

Answer 2

[tth_new]

Cách khác: \(A=\left(x-1\right)^2+\left(y-1\right)^2+3=\frac{1}{2}\left(x+y-2\right)^2+\frac{1}{2}\left(x-y\right)^2+3\ge3\)