`x ;y;z > 0 ; 4x^2 + 4y^2 + 17xy + 5x + 5y >= 1` Tìm Min `: P = 17x^2 + 17y^2 + 16xy `

`#081`

`+)` Do `x,y > 0` `=>` `xy ≤ ((x+y)^2)/4` `(`Bất đẳng thức Bunhiacopxki`)`

Đặt: `x+y = a>0`

`=>` `xy ≤ (a^2)/4` `(1)`

`+)` Ta có:

`1 ≤ 4x^2 + 4y^2 + 17xy + 5x + 5y`

`=` `4(x^2 +y^2 + 2xy) + 9y + 5 (x+y)`

`=` `4(x+y)^2 + 9xy + 5(x+y) ≤ 4a^2 + 9/4 a^2 + 5a`

`=` `(25a^2)/4 + 5a`

`=` `((25a^2)/4 + 5a + 1) - 1`

`=` `((5a)/2 + 1)^2 - 1`

`=>` `((5a)/2 + 1)^2 ≥ 2` `=>` `5a/2 + 1 ≥`$\sqrt{2}$ 

`=>` `5a ≥ -2 +``2`$\sqrt{2}$ `=>` `a≥ (-2 + 2\sqrt{2})/5`

`=>` `P = 17x^2 + 17y^2 + 16xy = 17(x^2 + y^2 +2xy) - 18xy`

`=` `17(x+y)^2 - 18xy ≥ 17a^2 - 18/4 a^2`

`=` `25/2 a^2 ≥ 25/2 ((-2 + 2\sqrt{2})/5)^2`

`=>` `P ≥ 6 -`$4\sqrt{2}$

Dấu " `=` " xảy ra `<=>` `x=y = 4/2 = ((-2 + 2\sqrt{2})/5)/2` `=` `(-1+sqrt{2})/5`

`=>` `Mi n _ (P) = 6-`$4\sqrt{2}$ khi `x=y = (-1+sqrt{2})/5`