Đáp án:
$\\$
`A = x^2 + 2x + 200`
`↔ A = x^2 + 2x + 1 + 199`
`↔ A = x^2 + 2x . 1 + 1^2 + 199`
`↔ A = (x+1)^2+199`
Với mọi `x` có : `(x+1)^2 ≥ 0`
`↔ (x+1)^2 + 199 ≥ 199 ∀x`
`↔ B ≥ 199 ∀x`
`↔ min B = 199`
Dấu "`=`" xảy ra khi :
`↔x+1=0`
`↔x=0-1`
`↔x=-1`
Vậy `min B=199 ↔ x=-1`
$\\$
`B = x^2 - x - 2021`
`↔ B = x^2 - 2 . 1/2x + 1/4 - 8085/4`
`↔ B = x^2 - 2 . 1/2x + (1/2)^2 - 8085/4`
`↔ B = (x-1/2)^2 - 8085/4`
Với mọi `x` có : `(x-1/2)^2 ≥ 0`
`↔ (x-1/2)^2 - 8085/4 ≥ (-8085)/4 ∀ x`
`↔ B ≥ (-8085)/4 ≥ x`
`↔ min B = (-8085)/4`
Dấu "`=`" xảy ra khi :
`↔x-1/2=0`
`↔x=0+1/2`
`↔x=1/2`
Vậy `min B=(-8085)/4 ↔ x=1/2`
$\\$
`C = 2x^2 + y^2 -2xy + 6x + 1`
`↔ C = x^2 + x^2 + y^2 -2xy + 6x + 9-8`
`↔ C = (x^2 - 2xy + y^2) + (x^2 + 6x + 9)-8`
`↔ C = (x-y)^2 + (x^2 + 2 . 3x + 3^2)-8`
`↔ C = (x-y)^2 + (x+3)^2-8`
Với mọi `x,y` có : \(\left\{ \begin{array}{l}(x-y)^2≥0\\(x+3)^2≥0\end{array} \right.\)
`↔ (x-y)^2 + (x+3)^2 ≥0∀x,y`
`↔ (x-y)^2 + (x+3)^2-8 ≥ 8 ∀ x,y`
`↔ C ≥ -8 ∀ x,y`
`↔ min C=-8`
Dấu "`=`" xảy ra khi :
`↔` \(\left\{ \begin{array}{l}x-y=0\\x+3=0\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}x=y\\x=-3\end{array} \right.\)
`↔` \(\left\{ \begin{array}{l}y=-3\\x=-3\end{array} \right.\)
Vậy `min C=-8 ↔ x=y=-3`
$\\$
`D = (x-3)^2 + (x-11)^2`
`↔ D = x^2 - 6x + 9+ x^2 - 22x + 121`
`↔ D = 2x^2 - 28x + 130`
`↔ D = 2 (x^2 - 14x + 65)`
`↔ D = 2 (x^2 - 2 . 7x + 49 + 16)`
`↔ D = 2 (x^2 - 2 . 7x + 7^2 + 16)`
`↔ D = 2 (x-7)^2 + 32`
Với mọi `x` có : `(x-7)^2 ≥ 0`
`↔ 2 (x-7)^2 ≥ 0 ∀x`
`↔ 2 (x-7)^2 + 32 ≥ 32 ∀x`
`↔ D≥32∀x`
`↔ min D=32`
Dấu "`=`" xảy ra khi :
`↔x-7=0`
`↔x=0+7`
`↔ x=7`
Vậy `min D=32 ↔ x=7`
