`a)4x-x²`
`=-(x²-4x)`
`=-(x²-4x+4-4)`
`=-(x²-4x+4)+4`
`=-(x²-2.x.2+2²)+4`
`=-(x-2)²+4`
Ta có:`(x-2)²≥0∀x`
`⇒-(x-2)²≤0∀x`
`⇒-(x-2)²+4≤4∀x`
Vậy `4x-x²_(max)=4` khi `x-2=0⇔x=2`
`b)6x-x²+1`
`=-(x²-6x-1)`
`=-(x²-6x+9-10)`
`=-(x²-6x+9)+10`
`=-(x²-2.x.3+3²)+10`
`=-(x-3)²+10`
Ta có:`(x-3)²≥0∀x`
`⇒-(x-3)²≤0∀x`
`⇒-(x-3)²+10≤10∀x`
Vậy `6x-x²+1_(max)=10` khi `x-3=0⇔x=3`
`c)5x-3x²+2`
`=-3(x²-5/3x-2/3)`
`=-3(x²-5/3x+25/36-49/36)`
`=-3(x²-5/3x+25/36)+49/12`
`=-3[x²-2.x. 5/6+(5/6)^2]+49/12`
`=-3(x-5/6)^2+49/12`
Ta có:`(x-5/6)^2≥0∀x`
`⇒3(x-5/6)^2≥0∀x`
`⇒-3(x-5/6)^2≤0∀x`
`⇒-3(x-5/6)^2+49/12≤49/12∀x`
Vậy `5x-3x²+2_(max)=49/12` khi `x-5/6=0⇔x=5/6`
`d)5x-x²+10`
`=-(x²-5x-10)`
`=-(x²-5x+25/4-65/4)`
`=-(x²-5x+25/4)+65/4`
`=-[x²-2.x. 5/2+(5/2)^2]+65/4`
`=-(x-5/2)^2+65/4`
Ta có:`(x-5/2)^2≥0∀x`
`⇒-(x-5/2)^2≤0∀x`
`⇒-(x-5/2)^2+65/4≤65/4∀x`
Vậy `5x-x²+10_(max)=65/4` khi `x-5/2=0⇔x=5/2`
`e)4-x²+2x`
`=-(x²-2x-4)`
`=-(x²-2x+1-5)`
`=-(x²-2x+1)+5`
`=-(x-1)²+5`
Ta có:`(x-1)²≥0∀x`
`⇒-(x-1)²≤0∀x`
`⇒-(x-1)²+5≤5∀x`
Vậy `4-x²+2x_(max)=5` khi `x-1=0⇔x=1`
`f)(1-3x)(x+2)`
`=x+2-3x²-6x`
`=-3x²-5x+2`
`=-3(x²+5/3x-2/3)`
`=-3(x²+5/3x+25/36-49/36)`
`=-3(x²+5/3x+25/36)+49/12`
`=-3[x²+2.x. 5/6+(5/6)^2]+49/12`
`=-3(x+5/6)^2+49/12`
Ta có:`(x+5/6)^2≥0∀x`
`⇒3(x+5/6)^2≥0∀x`
`⇒-3(x+5/6)^2≤0∀x`
`⇒-3(x+5/6)^2+49/12≤49/12∀x`
Vậy `(1-3x)(x+2)_(max)=49/12` khi `x+5/6=0⇔x=-5/6`
`g)x-x²-1`
`=-(x²-x+1)`
`=-(x²-x+1/4+3/4)`
`=-(x²-x+1/4)-3/4`
`=-[x²-2.x. 1/2+(1/2)^2]-3/4`
`=-(x-1/2)^2-3/4`
Ta có:`(x-1/2)^2≥0∀x`
`⇒-(x-1/2)^2≤0∀x`
`⇒-(x-1/2)^2-3/4≤-3/4∀x`
Vậy `x-x²-1_(max)=-3/4` khi `x-1/2=0⇔x=1/2`
`h)5-x²+2x-4y²-4y`
`=-1-1+7-x²+2x-4y²-4y`
`=-(x²-2x+1)-(4y²+4y+1)+7`
`=-(x²-2.x.1+1²)-[(2y)²+2.2y.1+1²]+7`
`=-(x-1)²-(2y+1)²+7`
Ta có:`(x-1)²≥0∀x`
`(2y+1)²≥0∀y`
`⇒-(x-1)²≤0∀x`
`-(2y+1)²≤0∀y`
`⇒-(x-1)²-(2y+1)²≤0∀x,y`
`⇒-(x-1)²-(2y+1)²+7≤7∀x,y`
Dấu `'='` xảy ra khi$\begin{cases} x-1=0\\2y+1=0 \end{cases}$`⇔`$\begin{cases} x=1\\y=-\dfrac{1}{2} \end{cases}$
Vậy `5-x²+2x-4y²-4y_(max)=7` khi `x=1` và `y=-1/2`
