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`1,`
`M = x^2 + 9x + 20`
`-> M =x^2 + 2 . x . 9/2 + (9/2)^2 -1/4`
`->M = (x+9/2)^2 - 1/4`
Vì `(x+9/2)^2 ≥0∀x`
`-> (x+9/2)^2 - 1/4 ≥ (-1)/4 ∀x`
`->M ≥ (-1)/4 ∀x`
Dấu "`=`" xảy ra khi :
`(x+9/2)^2=0 ↔ x+9/2=0 ↔x=(-9)/2`
Vậy `min M = (-1)/4 ↔x=(-9)/2`
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`2,`
`N = -8x + 2x^2 - 17`
`-> N =2x^2 - 8x - 17`
`->N = 2 (x^2 - 4x - 17/2)`
`->N = 2 (x^2 - 2 . x . 2 +2^2 - 25/2)`
`-> N = 2 (x-2)^2 - 25`
Vì `(x-2)^2 ≥0∀x`
`->2(x-2)^2 ≥0∀x`
`->2(x-2)^2 - 25 ≥ -25 ∀x`
`->N ≥ -25 ∀x`
Dấu "`=`" xảy ra khi :
`(x-2)^2=0 ↔ x-2=0 ↔x=2`
Vậy `min N=-25↔x=2`
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`3,`
`K = x^2 +2y^2 - 4y - 6x + 18`
`->K = (x^2 - 6x)+(2y^2 - 4y)+18`
`-> K = (x^2 - 2 . x . 3 + 3^2) + 2 (y^2 - 2 . y . 1 +1^2) + 7`
`->K = (x-3)^2 + 2(y-1)^2 + 7`
Vì `(x-3)^2 ≥0∀x, (y-1)^2 ≥0∀y`
`-> (x-3)^2 + 2(y-1)^2 + 7 ≥7∀x,y`
`->K ≥7∀x,y`
Dấu "`=`" xảy ra khi :
`(x-3)^2=0, (y-1)^2=0`
`↔ x-3=0,y-1=0`
`↔x=3,y=1`
Vậy `min K=7 ↔x=3,y=1`
