Bài 14:
`a)A=x²-20x+101`
`=x²-20x+100+1`
`=(x²-20x+100)+1`
`=(x²-2.x.10+10²)+1`
`=(x-10)²+1`
Ta có:`(x-10)²≥0∀x`
`⇒(x-10)²+1≥1∀x`
Vậy `A_(min)=1` khi `x-10=0⇔x=10`
`b)B=4x²+4x+2`
`=4x²+4x+1+1`
`=(4x²+4x+1)+1`
`=[(2x)²+2.2x.1+1²]+1`
`=(2x+1)²+1`
Ta có:`(2x+1)²≥0∀x`
`⇒(2x+1)²+1≥1∀x`
Vậy `B_(min)=1` khi `2x+1=0⇔x=-1/2`
`c)C=x²-4xy+5y²+10x-22y+28`
`=x²-4xy+4y²+y²+10x-20y-2y+1+25+2`
`=(x²-4xy+4y²)+(10x-20y)+25+(y²-2y+1)+2`
`=[x²-2.x.2y+(2y)²]+10(x-2y)+25+(y²-2.y.1+1²)+2`
`=(x-2y)²+2.(x-2y).5+5²+(y-1)²+2`
`=(x-2y+5)²+(y-1)²+2`
Ta có:`(x-2y+5)²≥0∀x,y`
`(y-1)²≥0∀y`
`⇒(x-2y+5)²+(y-1)²≥0∀x,y`
`⇒(x-2y+5)²+(y-1)²+2≥2∀x,y`
Dấu `'='` xảy ra khi$\begin{cases} x-2y+5=0\\y-1=0 \end{cases}$`⇔`$\begin{cases} x=-3\\y=1 \end{cases}$
Vậy `C_(min)=2` khi `x=-3` và `y=1`
`d)D=2x²-6x`
`=2(x²-3x)`
`=2(x²-3x+9/4-9/4)`
`=2(x²-3x+9/4)-9/2`
`=2[x²-2.x. 3/2+(3/2)^2]-9/2`
`=2(x-3/2)^2-9/2`
Ta có:`(x-3/2)^2≥0∀x`
`⇒2(x-3/2)^2≥0∀x`
`⇒2(x-3/2)^2-9/2≥-9/2∀x`
Vậy `D_(min)=-9/2` khi `x-3/2=0⇔x=3/2`
`e)E=x²+5x+7`
`=x²+5x+25/4+3/4`
`=(x²+5x+25/4)+3/4`
`=[x²+2.x. 5/2+(5/2)^2]+3/4`
`=(x+5/2)^2+3/4`
Ta có:`(x+5/2)^2≥0∀x`
`⇒(x+5/2)^2+3/4≥3/4∀x`
Vậy `E_(min)=3/4` khi `x+5/2=0⇔x=-5/2`
`g)G=x²+3x+7`
`=x²+3x+9/4+19/4`
`=(x²+3x+9/4)+19/4`
`=[x²+2.x. 3/2+(3/2)^2]+19/4`
`=(x+3/2)^2+19/4`
Ta có:`(x+3/2)^2≥0∀x`
`⇒(x+3/2)^2+19/4≥19/4∀x`
Vậy `G_(min)=19/4` khi `x+3/2=0⇔x=-3/2`
`h)H=x²+5x-6`
`=x²+5x+25/4-49/4`
`=(x²+5x+25/4)-49/4`
`=[x²+2.x. 5/2+(5/2)^2]-49/4`
`=(x+5/2)^2-49/4`
Ta có:`(x+5/2)^2≥0∀x`
`⇒(x+5/2)^2-49/4≥-49/4∀x`
Vậy `H_(min)=-49/4` khi `x+5/2=0⇔x=-5/2`
`i)G=2x²+3x+1`
`=2(x²+3/2x+1/2)`
`=2(x²+3/2x+9/16-1/16)`
`=2(x²+3/2x+9/16)-1/8`
`=2[x²+2.x. 3/4+(3/4)^2]-1/8`
`=2(x+3/4)^2-1/8`
Ta có:`(x+3/4)^2≥0∀x`
`⇒2(x+3/4)^2≥0∀x`
`⇒2(x+3/4)^2-1/8≥-1/8∀x`
Vậy `G_(min)=-1/8` khi `x+3/4=0⇔x=-3/4`
