A = x² - 2x + 9 = ( x² - 2x + 1 ) + 8 = ( x - 1 )² + 8 ≥ 8 ∀ x

Dấu "=" xảy ra khi x = 1

=> MinA = 8 <=> x = 1

B = x² + 6x - 3 = ( x² + 6x + 9 ) - 12 = ( x + 3 )² - 12 ≥ -12 ∀ x

Dấu "=" xảy ra khi x = -3

=> MinB = -12 <=> x = -3

C = ( x - 1 )( x - 3 ) + 9 = x² - 4x + 3 + 9 = ( x² - 4x + 4 ) + 8 = ( x - 2 )² + 8 ≥ 8 ∀ x

Dấu "=" xảy ra khi x = 2

=> MinC = 8 <=> x = 2

D = -x² - 4x + 7 = -( x² + 4x + 4 ) + 11 = -( x + 2 )² + 11 ≤ 11 ∀ x

Dấu "=" xảy ra khi x = -2

=> MaxD = 11 <=> x = -2

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\(A=\left(x-1\right)^2+8\ge8\\ A_{min}=8\Leftrightarrow x=1\\ B=\left(x+3\right)^2-12\ge-12\\ B_{min}=-12\Leftrightarrow x=-3\\ C=x^2-4x+3+9=\left(x-2\right)^2+8\ge8\\ C_{min}=8\Leftrightarrow x=2\\ E=-\left(x+2\right)^2+11\le11\\ E_{max}=11\Leftrightarrow x=-2\\ F=9-4x^2\le9\\ F_{max}=9\Leftrightarrow x=0\)

a, \(A=-x^2-2x+3=-\left(x^2+2x-3\right)=-\left(x^2+2x+1-4\right)\)

\(=-\left(x+1\right)^2+4\le4\)

Dấu ''='' xảy ra khi x = -1 

Vậy GTLN là 4 khi x = -1 

b, \(B=-4x^2+4x-3=-\left(4x^2-4x+3\right)=-\left(4x^2-4x+1+2\right)\)

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

Dấu ''='' xảy ra khi x = 1/2 

Vậy GTLN B là -2 khi x = 1/2 

c, \(C=-x^2+6x-15=-\left(x^2-2x+15\right)=-\left(x^2-2x+1+14\right)\)

\(=-\left(x-1\right)^2-14\le-14\)

Vâỵ GTLN C là -14 khi x = 1

Bài 8 : 

b, \(B=x^2-6x+11=x^2-6x+9+2=\left(x-3\right)^2+2\ge2\)

Dấu ''='' xảy ra khi x = 3

Vậy GTNN B là 2 khi x = 3 

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

Dấu ''='' xảy ra khi x = 1/2 

Vậy ...

c, \(x^2-12x+2=x^2-12x+36-34=\left(x-6\right)^2-34\ge-34\)

Dấu ''='' xảy ra khi x = 6

Vậy ...

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

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 "=" \(\Leftrightarrow x=2\)

\(A=\left[\frac{6x^2}{x^3-1}-\frac{2x-2}{x^2+x+1}-\frac{1}{x-1}\right]:\frac{x^2+9}{\left(x-1\right)\left(9-4x\right)}\)

\(=\left[\frac{6x^2}{x^3-1}-\frac{\left(2x-2\right)\left(x-1\right)}{\left(x^2+x+1\right)\left(x-1\right)}-\frac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\right]\cdot\frac{\left(x-1\right)\left(9-4x\right)}{x^2+9}\)

\(=\frac{6x^2-\left(2x^2-4x+2\right)-x^2-x-1}{\left(x^2+x+1\right)\left(x-1\right)}\cdot\frac{\left(x-1\right)\left(9-4x\right)}{x^2+9}\)

\(=\frac{5x^2-2x^2+4x-2-x-1}{\left(x^2+x+1\right)}\cdot\frac{\left(9-4x\right)}{x^2+9}\)

\(=\frac{3x^2+3x-3}{\left(x^2+x+1\right)}\cdot\frac{\left(9-4x\right)}{x^2+9}\)

Biểu thức A bạn viết đúng chưa?

A = x² + 4x + 7

   = ( x² + 4x + 4 ) + 3

   = ( x + 2 )² + 3

( x + 2 )² ≥ 0 ∀ x => ( x + 2 )² + 3 ≥ 3

Đẳng thức xảy ra <=> x + 2 = 0 => x = -2

=> MinA = 3 <=> x = -2

B = 2x² - 6x 

   = 2( x² - 3x + 9/4 ) - 9/2

   = 2( x - 3/2 )² - 9/2

2( x - 3/2 )² ≥ 0 ∀ x => 2( x - 3/2 )² -9/2 ≥ -9/2 

Đẳng thức xảy ra <=> x - 3/2 = 0 => x = 3/2

=> MinB = -9/2 <=> x = 3/2

C = -2x² + 8x - 15

    = -2( x² - 4x + 4 ) - 7

    = -2( x - 2 )² - 7

-2( x - 2 )² ≤ 0 ∀ x => -2( x - 2 )² - 7 ≤ -7

Đẳng thức xảy ra <=> x - 2 = 0 => x = 2

=> MaxC = -7 <=> x = 2

2. Ta có: A = x² - 6x + 5 = (x² - 6x + 9) - 4 = (x - 3)² - 4 

Ta luôn có: (x - 3)² \(\ge\)0 \(\forall\)x

=> (x - 3)² - 4 \(\ge\)-4 \(\forall\)x

Dấu "=" xảy ra <=> x - 3 = 0 <=> x = 3

Vậy MinA = -4 tại  x = 3

Ta có: B = 4x² - 8x + 7 = 4(x² - 2x + 1) + 3 = 4(x - 1)² + 3

Ta luôn có: 4(x - 1)² \(\ge\)0 \(\forall\)x

=> 4(x - 1)² + 3 \(\ge\)3 \(\forall\)x

Dấu "=" xảy ra <=> x - 1 = 0 <=> x = 1

vậy MinB = 3 tại x = 1

Ta có: C = 2x² + 4x - 6 = 2(x² + 2x + 1) - 8 = 2(x + 1)² - 8

Ta luôn có: 2(x + 1)² \(\ge\)0 \(\forall\)x

=> 2(x + 1)² - 8 \(\ge\)-8 \(\forall\)x

Dấu "=" xảy ra <=> x + 1 = 0 <=> x = -1

Vậy MinC = -8 tại x = -1

1/

\(A=x^2-6x+5\)

\(A=x^2-2\cdot3x+3^2-3^2+5\)

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

\(A=\left(x-3\right)^2-9+5\)

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

mà \(\left(x-3\right)^2\ge0\Rightarrow\left(x-3\right)^2-4\ge-4\)

\(\Rightarrow GTNNA\left(x^2-6x+5\right)=-4\)

với \(\left(x-3\right)^2=0;x=3\)

\(B=4x^2-8x+7\)

\(B=4\left(x^2-2x+\frac{7}{4}\right)\)

\(B=4\left(x^2-2\cdot1x+1-1+\frac{7}{4}\right)\)

\(B=4\left(x-1\right)^2+3\)

\(\left(x-1\right)^2\ge0\Rightarrow4\left(x^2-1\right)^2+3\ge3\)

\(\Rightarrow GTNNB=3\)

với \(\left(x-1\right)^2=0;x=1\)

\(C=2x^2+4x-6\)

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

\(C=2\left(x^2+2\cdot1x+1-1-3\right)\)

\(C=\left(x+1\right)^2-8\)

có\(\left(x+1\right)^2\ge0\Rightarrow\left(x+1\right)^2-8\ge-8\)

\(\Rightarrow GTNNC=-8\)

với \(\left(x+1\right)^2=0;x=-1\)

3

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

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

\(A=-\left(x^2-2\cdot1x+1-1+3\right)\)

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

có \(\left(x-1\right)^2\ge0\Rightarrow\left(x-1\right)^2-2\ge-2\)

\(\Rightarrow GTLNA=-2\) với \(\left(x-1\right)^2=0;x=1\)

\(B=-9x^2+6x-4\)

\(B=-9\left(x^2-\frac{2}{3}x+\frac{4}{9}\right)\)

\(B=-9\left(x^2-2\cdot\frac{1}{3}x+\frac{1}{9}-\frac{1}{9}+\frac{4}{9}\right)\)

\(B=-9\left(x^2-2\cdot\frac{1}{3}x+\frac{1}{9}-\frac{3}{9}\right)\)

\(B=-9\left(x-\frac{1}{3}\right)^2+3\)

có \(\left(x-\frac{1}{3}\right)^2\ge0\Rightarrow-9\left(x-\frac{1}{3}\right)^2+3\ge3\)

\(\Rightarrow GTLNB=3\)với \(\left(x-\frac{1}{3}^2=0;\right)x=\frac{1}{3}\)

2.

c) \(C=2x^2+4x-6=2\left(x^2+2x+1\right)-8\)

\(=2\left(x+1\right)^2-8\ge-8\forall x\)

Dấu"=" xảy ra<=> \(2\left(x+1\right)^2=0\Leftrightarrow x=-1\)

3.

c) \(C=-3x^2-6x+9=-3\left(x^2+2x+1\right)+12\)

\(=-3\left(x+1\right)^2+12\le12\forall x\)

Dấu "=" xảy ra<=> \(-3\left(x+1\right)^2=0\Leftrightarrow x=-1\)

\(2,GTNN\)

\(A=x^2-6x+5=x^2+6x+9-4\)

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

\(A_{min}=-4\Leftrightarrow\left(x+3\right)^2=0\Rightarrow x=-3\)

\(B=4x^2-8x+7=4\left(x^2-2x+\frac{7}{4}\right)\)

\(=4\left(x^2-2x+1+\frac{3}{4}\right)=4\left(x-1\right)^2+3\ge3\)

\(\Rightarrow B_{min}=3\Leftrightarrow\left(x-1\right)^2=0\Rightarrow x=1\)

\(C=2x^2+4x-6=2\left(x^2+2x-3\right)\)

\(=2\left(x^2+2x+1-4\right)=2\left(x+1\right)^2-8\ge-8\)

\(\Rightarrow C_{min}=-8\Leftrightarrow\left(x+1\right)^2=0\Rightarrow x=-1\)

\(3,GTLN\)

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

\(=-\left(x^2-2x+1-4\right)=-\left(x-1\right)^2+4\le4\)

\(A_{max}=4\Leftrightarrow-\left(x-1\right)^2=0\Rightarrow x=1\)

\(B=-9x^2+6x-4=-\left[9x^2-6x+4\right]\)

\(=-\left[\left(3x\right)^2-6x+1+3\right]=-\left(3x-1\right)^2-3\)

\(B_{max}=-3\Leftrightarrow-\left(3x-1\right)^2=0\Rightarrow x=\frac{1}{3}\)

\(C=-3x^2-6x+9=-3\left(x^2+2x-3\right)\)

\(=-3\left(x^2+2x+1-4\right)=-3\left(x+1\right)^2+12\)

\(C_{max}=12\Leftrightarrow-3\left(x+1\right)^2=0\Rightarrow x=-1\)

2/

A= X²-6X+5

  = X²-2.3.X+3²-4

 = (X-3)²-4 ( giá trị nhỏ nhất là -4)

B=4X²-8X+7

 =(2X)²-2.2.2X+2²+3

 =(2X-2)²+3 ( giá trị nhỏ nhất là 3 )

C= 2X²+ 4X-6

  = 2(X²+2X+1²-7)

  = 2((X+1)²-7)

  = 2(X+1)²-14 ( giá trị nhỏ nhất là -14 )

bài 1:

b, x² - 6x +10=x² - 2.x.3 +9 +1=(x - 3)² +1

Vì (x-3)²  >= 0 với mọi x

=> (x-3)² +1 >= 1 với mọi x

vậy GTNN của biểu thức bằng 1 <=> x-3=0<=> x=3