`A = x^2 + 4x + 6`
`= x^2 + 2*2x + 4 + 2`
`= ( x + 2 )^2 + 2`
Do `( x + 2 )^2` $\geqslant$ `0`
`=> ( x + 2 )^2 + 2` $\geqslant$ `2`
Dấu ''='' xảy ra `:`
`( x + 2 )^2 + 2` $=$ `2`
`<=> ( x + 2 )^2 = 0`
`<=> x + 2 = 0`
`<=> x = -2`
Vậy $Min$ `A = 2 <=> x = -2`
`B = x^2 - 6x + 4`
`= x^2 - 2*3x + 9 - 5`
`= ( x-3 )^2 - 5`
Do `( x -3)^2` $\geqslant$ `0`
`=> ( x-3 )^2 - 5` $\geqslant$ `-5`
Dấu ''='' xảy ra `:`
`( x-3 )^2 - 5` $=$ `-5`
`<=> ( x-3 )^2 = 0`
`<=> x-3 = 0`
`<=> x = 3`
Vậy $Min$ `B = -5 <=> x=3`
`C = -( x^2 + 2x + 3 )`
`= -( x^2 - 2x - 3 )`
`= -( x^2 - 2x + 1 - 4 )`
`= 4 - ( x - 1 )^2`
Do `( x - 1 )^2` $\geqslant$ `0`
`=> - ( x - 1 )^2` $\leqslant$ `0`
`=> 4 - ( x - 1 )^2` $\leqslant$ `4`
Dấu ''='' xảy ra `:`
`4 - ( x - 1 )^2` `=` `4`
`<=> ( x-1 )^2 = 0`
`<=> x-1=0`
`<=> x=1`
Vậy $Max$ `C = 4 <=> x=1`
`D = 2x^2 + 3x`
`= 2( x^2 + 3/2x )`
`= 2( x^2 + 3/2x + 9/36 ) - 9/8`
`= 2( x + 3/2 )^2 - 9/8`
Do `2( x + 3/2 )^2` $\geqslant$ `0`
`=> 2( x + 3/2 )^2 - 9/8` $\geqslant$ `-9/8`
Dấu ''='' xảy ra `:`
`2( x + 3/2 )^2 - 9/8` `=` `-9/8`
`<=> 2( x + 3/2 )^2 =0`
`<=> ( x + 3/2 )^2 =0`
`<=> x+3/2=0`
`<=> x=-3/2`
Vậy $Min$ `D = -9/8 <=> x = -3/2`
`E = x^2 + y^2 - x + 6y + 10`
`= ( x^2 - x + 1/4 ) + ( y^2 + 6x + 9 ) + 3/4`
`= ( x-1/2 )^2 + ( y+3)^2 + 3/4`
Do `{(( x-1/2 )^2 ⩾ 0 ),(( y+3)^2 ⩾ 0 ):}`
`=> ( x-1/2 )^2 + ( y+3)^2 ⩾ 0`
`=> ( x-1/2 )^2 + ( y+3)^2 + 3/4` `⩾ 3/4`
Dấu ''='' xảy ra `:`
`( x-1/2 )^2 + ( y+3)^2 + 3/4` `= 3/4`
`<=> ( x-1/2 )^2 + ( y+3)^2 = 0`
`<=> {(( x-1/2 )^2 = 0 ),(( y+3)^2 =0 ):}`
`<=> {( x-1/2 = 0 ),( y+3 =0 ):}`
`<=> {( x = 1/2 ),( y =-3 ):}`
Vậy $Min$ `E = 3/4 <=> x = 1/2 ; y = -3`
`F = 2x - 2x^2 - 5`
`= -( 2x^2 - 2x + 5 )`
`= -2( x^2 - x + 5/2 )`
`= -2( x^2 - x + 1/4 + 9/4 )`
`= -2[(x-1/2)^2 + 9/4]`
Do `(x-1/2)^2 ⩾ 0`
`=> (x-1/2)^2 + 9/4 ⩾ 9/4`
`=> -2[(x-1/2)^2 + 9/4]` $\leqslant$ `-9/2`
Dấu ''='' xảy ra `:`
` -2[(x-1/2)^2 + 9/4]` `=` `-9/2`
`<=> ( x-1/2)^2 = 0`
`<=> x-1/2=0`
`<=> x= 1/2`
Vậy $Max$ `F = -9/2 <=> x=1/2`
