Question

tim MIN  A = x^2+y^2-xy-x+y+1

Answer 1

2.A = 2x² + 2y² - 2xy  - 2x + 2y + 2 = (x² - 2xy + y² ) + (x² - 2x + 1) + (y² + 2y + 1) = (x - y)² + (x - 1)² + (y +1)²

= (x - y)² + (1 - x)² + (y +1)²

Ap dụng bđt Bu nhi a: (ax + by + cz)² \(\le\) (a² + b² + c²)(x² + y² + z²). dấu = xảy ra khi a/x = b/y = c/z

ta có [(x - y).1 + (1- x).1 + (y + 1).1]² \(\le\) [(x - y)² + (1 - x)² + (y +1)²].(1² + 1² + 1²)

=> 4 \(\le\) 3. 2.A => A \(\ge\)2/3 => Min A = 2/3

dấu = xảy ra khi x - y = 1- x = y + 1 => x = 1/3; y = -1/3