Đáp án:
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`a,`
`P = x^2 - 6x + 11`
`⇔ P = x^2 - 2 . 3x + 9 + 2`
`⇔ P = x^2 - 2 . 3x + 3^2 + 2`
`⇔ P = (x-3)^2 + 2`
Với mọi `x` có : `(x-3)^2 ≥ 0`
`⇔ (x-3)^2 + 2 ≥ 2 ∀x`
`⇔ P ≥ 2∀X`
`⇔ min P=2`
Dấu "`=`" xảy ra khi :
`⇔x-3=0`
`⇔x=0+3`
`⇔x=3`
Vậy `min P=2⇔x=3`
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`b,`
`Q = y^2 + y`
`⇔ Q = y^2 + y + 1/4 - 1/4`
`⇔ Q = y^2 + 2 . 1/2y + (1/2)^2 - 1/4`
`⇔ Q = (y + 1/2)^2 - 1/4`
Với mọi `y` có : `(x+1/2)^2 ≥0`
`⇔ (x+1/2)^2 - 1/4 ≥ (-1)/4 ∀ y`
`⇔ Q ≥ (-1)/4 ∀ y`
`⇔min Q = (-1)/4`
Dấu "`=`" xảy ra khi :
`⇔x+1/2=0`
`⇔x=0-1/2`
`⇔x=(-1)/2`
Vậy `min Q=(-1)/4 ⇔ x=(-1)/2`
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`c,`
`K = x^2 + y^2 - 6x + y + 10`
`⇔ K = x^2 + y^2 - 6x + y + 9 + 1`
`⇔ K = [x^2 - 6x + 9] + [y^2 + y + 1]`
`⇔ K = [x^2 - 2 . 3x + 3^2] + [y^2 + 2 . 1/2y + 1/4 + 3/4]`
`⇔ K = (x-3)^2 + [y^2 + 2 . 1/2y + (1/2)^2 + 3/4]`
`⇔ K = (x-3)^2 + (y+1/2)^2 + 3/4`
Với mọi `x,y` có : `(x-3)^2 ≥ 0, (y+1/2)^2 ≥ 0`
`⇔ (x-3)^2 + (y+1/2)^2 ≥ 0 ∀ x,y`
`⇔ (x-3)^2 + (y+1/2)^2 + 3/4 ≥ 3/4 ∀ x,y`
`⇔ K ≥ 3/4 ∀ x,y`
`⇔ min K=3/4`
Dấu "`=`" xảy ra khi :
`⇔x-3=0,y+1/2=0`
`⇔x=0+3,y=0-1/2`
`⇔x=3,y=(-1)/2`
Vậy `min K=3/4 ⇔ x=3,y=(-1)/2`
