`d)`
`D=4x^2-4x`
`=4x^2-4x+1-1`
`=(2x)^2-2.2x+1-1`
`=(2x-1)^2-1`
Vì `(2x-1)^2>=0∀x`
`->(2x-1)^2-1>=-1∀x`
`->D>=-1`
Dấu `'='` xảy ra `<=>2x-1=0<=>x=1/2`
Vậy `D_{min}=-1` khi `x=1/2`
`e)`
`E=5-8x+x^2`
`=x^2-8x+16-11`
`=x^2-2.x.4+4^2-11`
`=(x-4)^2-11`
Vì `(x-4)^2≥0∀x`
`->(x-4)^2-11≥-11∀x`
`->E≥-11∀x`
Dấu `'='` xảy ra `<=>x-4=0<=>x=4`
Vậy `E_{min}=-11` khi `x=4`
`g)`
`F=(x-1)(x+2)(x+3)(x+6)`
`=[(x-1).(x+6)].[(x+2).(x+3)]`
`=(x^2-6x+x-6).(x^2+2x+3x+6)`
`=(x^2+5x-6).(x^2+5x+6)`
`=(x^2+5x)^2-6^2`
`=(x^2+5x)^2-36`
Vì `(x^2+5x)^2>=0∀x`
`->(x^2+5x)^2-36≥-36∀x`
`->F≥-36`
Dấu `'='` xảy ra `<=>x^2+5x=0`
`<=>x(x+5)=0`
`<=>`\(\left[ \begin{array}{l}x=0\\x+5=0\end{array} \right.\)
`<=>`\(\left[ \begin{array}{l}x=0\\x=-5\end{array} \right.\)
Vậy `F_{min}=-36` khi `x∈{-5;0}`
`h)`
`H=x^2+5y^2-2xy+4y+3`
`=x^2-2xy+y^2+4y^2+4y+4+1+2`
`=(x-y)^2+(2y+1)^2+2`
Vì `(x-y)^2>=0∀x;y`
`(2y+1)^2>=0∀y`
`->(x-y)^2+(2y+1)^2>=0∀x;y`
`->(x-y)^2+(2y+1)^2+2>=2∀x;y`
`->H>=2`
Dấu `'='` xảy ra `<=>{(x-y=0),(2y+1=0):}`
`<=>{(x=y),(y=-1/2):}`
`<=>{(x=-1/2),(y=-1/2):}`
Vậy `H_{min}=2` khi `(x;y)=(-1/2;-1/2)`
`k)`
`K=(x^2-2x)(x^2-2x+2)`
`=(x^2-2x+1-1).(x^2-2x+1+1)`
`=[(x-1)^2-1].[(x-1)^2+1]`
`=[(x-1)^2)^2-1`
`=(x-1)^4-1`
Vì `(x-1)^4>=0∀x`
`->(x-1)^4-1≥-1∀x`
`->K≥-1`
Dấu `'='` xảy ra `<=>x-1=0`
`<=>x=1`
Vậy `K_{min}=-1` khi `x=1`
`m)`
`M=(x+1).(2x-1)`
`=2x^2-x+2x-1`
`=2x^2+x-1`
`=2.(x^2+1/2 x )-1`
`=2[x^2 +2. 1/4 x +(1/4)^2-(1/4)^2]-1`
`=2.[(x+1/4)^2-1/16]-1`
`=2(x+1/4)^2-1/8-1`
`=2(x+1/4)^2-9/8`
Vì `(x+1/4)^2≥0∀x`
`->2(x+1/4)^2≥0∀x`
`->2(x+1/4)^2-9/8≥-9/8∀x`
`->M≥-9/8`
Dấu `'='` xảy ra `<=>x+1/4=0`
`<=>x=-1/4`
Vậy `M_{min}=-9/8` khi `x=-1/4`
`n)`
`N=4x^2-4xy+2y^2+1`
`=(2x)^2-2.2x.y+y^2+y^2+1`
`=(2x-y)^2+y^2+1`
Vì `(2x-y)^2≥0∀x;y`
`y^2≥0∀y`
`->(2x-y)^2+y^2≥0∀x;y`
`->(2x-y)^2+y^2+1≥1∀x;y`
`->N≥1`
Dấu `'='` xảy ra `<=>{(2x-y=0),(y=0):}`
`<=>{(2x=y),(y=0):}`
`<=>{(x=0),(y=0):}`
Vậy `N_{min}=1` khi `(x;y)=(0;0)`
