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$\bullet$ $\text{ Tìm max }$
`A = 6x - x^2`
`= -( x^2 - 6x )`
`= -( x^2 - 6x + 9 ) + 9`
`= -( x - 3 )^2 + 9`
$\text{ Do }$ `-( x - 3 )^2 ≤ 0 ∀x`
`⇒ -( x - 3 )^2 + 9` `≤9`
$\text{ Dấu ''='' xảy ra: }$
`-( x - 3 )^2 + 9` `=9`
`⇔ -( x - 3 )^2 = 0`
`⇔ x - 3 = 0`
`⇔ x = 3`
$\text{ Vậy }$ `A_max = 9 ⇔ x = 3`
`B = 6 - 8x - x^2`
`= -( x^2 + 8x - 6 )`
`= -( x^2 + 8x + 16 - 22 )`
`= -( x + 4 )^2 + 22`
$\text{ Do }$ `-( x + 4 )^2 ≤ 0 ∀x`
`⇒ -( x + 4 )^2 + 22` `≤22`
$\text{ Dấu ''='' xảy ra: }$
`-( x + 4 )^2 + 22` `≤22`
`<=> ( x + 4 )^2 = 0`
`<=> x + 4 = 0`
`<=> x = -4`
$\text{ Vậy }$ `B_max = 22 ⇔ x = -4`
`C = 2/( 2x^2 + 4x + 9 )`
`= 1/( x^2 + 2x + 9/2 )`
`= 1/[( x + 1 )^2 + 7/2 ]`
$\text{ Do }$ `( x + 1 )^2 >= 0`
`=> ( x + 1 )^2+7/2 >= 7/2`
`=> 1/[( x + 1 )^2 + 7/2 ]` `<= 2/7`
$\text{ Dấu ''='' xảy ra: }$
`1/[( x + 1 )^2 + 7/2 ]` `= 2/7`
`<=> ( x + 1 )^2 + 7/2 = 7/2`
`<=> ( x + 1 )^2 = 0`
`<=> x + 1 = 0`
`<=> x = -1`
$\text{ Vậy }$ `C_max = 2/7 ⇔ x = -1`
$\bullet$ $\text{ Tìm min }$
`C = x^2 - 4x + 7`
`= x^2 - 4x + 4 + 3`
`= ( x-2 )^2 + 3`
$\text{ Do }$ `( x-2 )^2 >= 0`
`=> ( x-2 )^2 + 3 >= 3`
$\text{ Dấu ''='' xảy ra: }$
`( x-2 )^2 + 3 = 3`
`<=> ( x-2)^2=0`
`<=> x-2=0`
`<=>x=2`
$\text{ Vậy }$ `C_min = 3 ⇔ x = 2`
`D = 4x^2 + 4x + 3`
`= 4x^2 + 4x + 1 + 2`
`= ( 2x + 1 )^2 + 2`
$\text{ Do }$ `( 2x+1 )^2 >= 0`
`=> ( 2x + 1 )^2 + 2` `>= 2`
$\text{ Dấu ''='' xảy ra: }$
`( 2x + 1 )^2 + 2` `= 2`
`<=> ( 2x+1)^2=0`
`<=> 2x+1=0`
`<=> x = -1/2`
$\text{ Vậy }$ `D_min = 2 ⇔ x = -1/2`
`E = x^2 + 9y^2 + 6x - 6y + 5`
`= ( x^2 + 6x + 9 ) + ( 9y^2 - 6y + 1 ) - 8`
`= ( x +3 )^2 + ( 3y - 1 )^2 - 5`
$\text{ Do }$ `{(( x + 3 )^2 >= 0 ),(( 3y-1 )^2 >= 0 ):}`
`=> ( x +3 )^2 + ( 3y - 1 )^2 >= 0`
`=> ( x +3 )^2 + ( 3y - 1 )^2 - 5` `>= -5`
$\text{ Dấu ''='' xảy ra: }$
`( x +3 )^2 + ( 3y - 1 )^2 - 5` `= -5`
`<=> ( x +3 )^2 + ( 3y - 1 )^2 = 0`
`<=> {(( x + 3 )^2 = 0 ),(( 3y-1 )^2 = 0 ):}`
`<=> {( x + 3 = 0 ),( 3y-1 = 0 ):}`
`<=> {( x = -3 ),( y = 1/3 ):}`
$\text{ Vậy }$ `E_min = -5 ⇔ x = -3;y=1/3`
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