Bài `1:`

`a`

`A = x^2 + 2y^2 - 2xy - 4y + 5`

`A = (x^2 - 2xy + y^2) + (y^2 - 4y + 4) + 1`

`A = (x - y)^2 + (y - 2)^2 + 1`

Vì `(x - y)^2 >= 0 AA x, y` 

`(y - 2)^2 >= 0 AA y`

`to A >= 1`

Dấu "`=`" xảy ra khi:

`{(x - y = 0),(y - 2 = 0):} to {(x = 2),(y = 2):}`

Vậy `min A = 1` khi `x = 2, y = 2.`

`b`

`B = 2x^2 - 4xy + 5y^2 + 5`

`B = 2(x^2 - 2xy + y^2) + 3y^2 + 5`

`B = 2(x - y)^2 + 3y^2 + 5`

Vì ` 2(x - y)^2 >= 0 AA x, y`

` 3y^2 >= 0 AA y`

`to B >= 5`

Dấu "`=`" xảy ra khi:

`{(x - y = 0),(y = 0):} to {(x = 0),(y = 0):}`

Vậy `min B = 5` khi `x = 0, y = 0.`

Bài `2:`

`a. A(x) = 2x^2 + y^2 - 2xy - 2x + 3`

`A(x) = (x^2 - 2xy + y^2) + (x^2 - 2x + 1) + 2`

`A(x) = (x - y)^2 + (x - 1)^2 + 2`

Vì `(x - y)^2 >= 0 AA x, y `

`(x - 1)^2 >= 0 AA x`

`to A(x) >= 2`

Dấu "`=`" xảy ra khi:

`{(x - y = 0),(x - 1 = 0):} to {(x = 1),(y = 1):}`

Vậy `min A(x) = 2` khi `x = 1, y = 1.`