c: \(-x^2+2x-2=-\left(x-1\right)^2-1\le-1\forall x\)

\(\Leftrightarrow V\ge-1\forall x\)

Dấu '=' xảy ra khi x=1

\(A=\left(x^2-2x+1\right)+4=\left(x-1\right)^2+4\ge4\\ A_{min}=4\Leftrightarrow x=1\\ B=2\left(x^2-3x\right)=2\left(x^2-2\cdot\dfrac{3}{2}x+\dfrac{9}{4}\right)-\dfrac{9}{2}\\ B=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\ge-\dfrac{9}{2}\\ B_{min}=-\dfrac{9}{2}\Leftrightarrow x=\dfrac{3}{2}\\ C=-\left(x^2-4x+4\right)+7=-\left(x-2\right)^2+7\le7\\ C_{max}=7\Leftrightarrow x=2\)

a,\(A=x^2-2x+5=\left(x^2-2x+1\right)+4=\left(x-1\right)^2+4\ge4\)

Dấu "=" \(\Leftrightarrow x=-1\)

b,\(B=2\left(x^2-3x\right)=2\left(x^2-3x+\dfrac{9}{4}\right)-\dfrac{9}{2}=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\ge-\dfrac{9}{2}\)

Dấu "=" \(\Leftrightarrow x=\dfrac{3}{2}\)

c,\(=C=-\left(x^2-4x-3\right)=-\left[\left(x^2-4x+4\right)-7\right]=-\left(x-2\right)^2+7\le7\)

Dấu "=" \(\Leftrightarrow x=2\)

a: A=(x-1)(x-3)(x²-4x+5)

\(=\left(x^2-4x+3\right)\left(x^2-4x+5\right)\)

\(=\left(x^2-4x\right)^2+8\left(x^2-4x\right)+15\)

\(=\left(x^2-4x+4\right)^2-1\)

\(=\left(x-2\right)^4-1>=-1\)

Dấu = xảy ra khi x-2=0

=>x=2

b: \(B=x^2-2xy+2y^2-2y+1\)

\(=x^2-2xy+y^2+y^2-2y+1\)

\(=\left(x-y\right)^2+\left(y-1\right)^2>=0\)

Dấu = xảy ra khi x-y=0 và y-1=0

=>x=y=1

c: \(C=5+\left(1-x\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\)

\(=-\left(x-1\right)\left(x+6\right)\left(x+2\right)\left(x+3\right)+5\)

\(=-\left(x^2+5x-6\right)\left(x^2+5x+6\right)+5\)

\(=-\left[\left(x^2+5x\right)^2-36\right]+5\)

\(=-\left(x^2+5x\right)^2+36+5\)

\(=-\left(x^2+5x\right)^2+41< =41\)

Dấu = xảy ra khi \(x^2+5x=0\)

=>x(x+5)=0

=>\(\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)

a: Ta có: \(x^2+x+1\)

\(=x^2+2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}\)

\(=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\forall x\)

Dấu '=' xảy ra khi \(x=-\dfrac{1}{2}\)

b: Ta có: \(-x^2+x+2\)

\(=-\left(x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{9}{4}\right)\)

\(=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{9}{4}\le\dfrac{9}{4}\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{1}{2}\)

1) \(A=x^2-4x+1\)

\(A=x^2-4x+4-3\)

\(A=\left(x^2-4x+4\right)-3\)

\(A=\left(x-2\right)^2-3\)

Ta có: \(\left(x-2\right)^2\ge0\) với mọi x

\(\Rightarrow\left(x-2\right)^2-3\ge-3\) với mọi x

Vậy MIinA = -3 khi x = 2

2) \(B=-x^2+13x+2012\)

\(B=-x^2+13x-\frac{169}{4}+\frac{169}{4}+2012\)

\(B=-\left(x^2-13+\frac{169}{4}\right)+\left(\frac{169}{4}+2012\right)\)

\(B=-\left(x-\frac{13}{2}\right)^2+\frac{8217}{4}\)

Ta có: \(\left(x-\frac{13}{2}\right)^2\ge0\) với mọi x

\(-\left(x-\frac{13}{2}\right)^2\le0\) với mọi x

\(\Rightarrow-\left(x-\frac{13}{2}\right)^2+\frac{8217}{4}\le\frac{8217}{4}\)

Vây \(Max\left(B\right)=\frac{8217}{4}\) khi \(x=\frac{13}{2}\)

\(A=\left(x-1\right)\left(x+3\right)\left(x+2\right)\left(x+6\right)=\left(x^2+5x-6\right)\left(x^2+5x+6\right)=\left(x^2+5x\right)^2-36\ge-36\)

\(minA=-56\Leftrightarrow x^2+5x=0\Leftrightarrow x\left(x+5\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)

\(B=4x-x^2+1=-\left(x^2-4x+4\right)+5=-\left(x-2\right)^2+5\le5\)

\(maxB=5\Leftrightarrow x=2\)

MinA=0

⇔x=1 hoặc x=-3 hoặc x=-2 hặc x=-6

B\(=-x^2+2x+1+2x\)

\(=-\left(x^2-2x+1\right)+2\left(1+x\right)\)

\(=-\left(x-1\right)^2-2\left(x-1\right)\)

`A=x^2-4x+1`
`=x^2-4x+4-3`
`=(x-2)^2-3>=-3`
Dấu "=" xảy ra khi x=2
`B=4x^2+4x+11`
`=4x^2+4x+1+10`
`=(2x+1)^2+10>=10`
Dấu "=" xảy ra khi `x=-1/2`
`C=(x-1)(x+3)(x+2)(x+6)`
`=[(x-1)(x+6)][(x+3)(x+2)]`
`=(x^2+5x-6)(x^2+5x+6)`
`=(x^2+5x)^2-36>=-36`
Dấu "=" xảy ra khi `x=0\or\x=-5`
`D=5-8x-x^2`
`=21-16-8x-x^2`
`=21-(x^2+8x+16)`
`=21-(x+4)^2<=21`
Dấu "=" xảy ra khi `x=-4`
`E=4x-x^2+1`
`=5-4+4-x^2`
`=5-(x^2-4x+4)`
`=5-(x-2)^2<=5`
Dấu "=" xảy ra khi `x=5`

A= x² - 4x +1

   = x² - 4x + 4 - 3

   = (x-2)² -3

Ta có (x-2)² ≥ 0 ∀ x

    ⇒ (x-2)² -3 ≥ -3 ∀ x

Vậy A_Min= -3 tại x=2

B= 4x²+4x+11

  = 4x²+4x+1+10

  = (2x+1)²+10

Ta có (2x+1)² ≥ 0 ∀ x

     ⇒ (2x+1)²+10 ≥ 10 ∀ x

Vậy B_Min=10 tại x= \(\dfrac{-1}{2}\)

C=(x-1)(x+3)(x+2)(x+6)

  = (x-1)(x+6)(x+3)(x+2)

  = (x²+5x-6) (x²+5x+6)

  = (x²+5x)² -36

Ta có (x²+5x)² ≥ 0 ∀ x
  ⇒ (x²+5x)² -36 ≥ -36 ∀ x

Vậy C_Min=-36 tại x=0 hoặc x= -5