\(A=x^2-6x+10\)

\(\Leftrightarrow A=x^2-2\cdot x\cdot3+3^2-9+10\)

\(\Leftrightarrow A=\left(x-3\right)^2+1\ge1\)     \(\forall x\in z\)

\(\Leftrightarrow A_{min}=1khix=3\)

\(B=3x^2-12x+1\)

\(\Leftrightarrow B=\left(\sqrt{3}x\right)^2-2\cdot\sqrt{3}x\cdot2\sqrt{3}+\left(2\sqrt{3}\right)^2-12+1\)

\(\Leftrightarrow B=\left(\sqrt{3}x-2\sqrt{3}\right)^2-11\ge-11\)    \(\forall x\in z\)

\(\Leftrightarrow B_{min}=-11khix=2\)

a) Ta có: \(Q=-x^2-y^2+4x-4y+2=-\left(x^2+y^2-4x+4y-2\right)\)

\(=-\left(x^2-4x+4+y^2+4y+4\right)+10\)

\(=-\left[\left(x-2\right)^2+\left(y+2\right)^2\right]+10\le10\forall x,y\)

Vậy Max_Q=10 khi x=2, y=-2

b) +Ta có: \(A=-x^2-6x+5=-\left(x^2+6x-5\right)=-\left(x^2+6x+9-14\right)\)

\(=-\left(x^2+6x+9\right)+14=-\left(x+3\right)^2+14\le14\forall x\)

Vậy Max_A=14 khi x=-3

+Ta có: \(B=-4x^2-9y^2-4x+6y+3=-\left(4x^2+9y^2+4x-6y-3\right)\)

\(=-\left(4x^2+4x+1+9y^2-6y+1-5\right)\)

\(=-\left[\left(2x+1\right)^2+\left(3y-1\right)^2\right]+5\le5\forall x,y\)

Vậy Max_B=5 khi x=-1/2, y=1/3

c) Ta có: \(P=x^2+y^2-2x+6y+12=x^2-2x+1+y^2+6y+9+2\)

\(=\left(x-1\right)^2+\left(y+3\right)^2+2\ge2\forall x,y\)

Vậy Min_P=2 khi x=1, y=-3

Ta có : P = x² - 2x + 5 = x² - 2x + 1 + 4 = (x - 1)² + 4

Vì \(\left(x-1\right)^2\ge0\forall x\)

Suy ra : \(P=\left(x-1\right)^2+4\ge4\forall x\)

Nên : P_min = 4 khi x = 1

b) Ta có Q = 2x² - 6x = 2(x² - 3x) = 2(x² - 3x + \(\frac{9}{4}-\frac{9}{4}\) ) = \(2\left(x^2-3x+\frac{9}{4}\right)-\frac{9}{2}=2\left(x-\frac{3}{2}\right)^2-\frac{9}{2}\)

Vì \(2\left(x-\frac{3}{2}\right)^2\ge0\forall x\) 

SUy ra ; \(Q=2\left(x-\frac{3}{2}\right)^2-\frac{9}{2}\ge-\frac{9}{2}\)

Vậy \(Q_{min}=-\frac{9}{2}\) khi \(x=\frac{3}{2}\)

Bài 1 : 

a, \(A=x^2-4x+6=x^2-4x+4+2=\left(x-2\right)^2+2\ge2\)

Dấu ''='' xảy ra khi x = 2 

Vậy GTNN A là 2 khi x = 2 

b, \(B=y^2-y+1=y^2-2.\frac{1}{2}y+\frac{1}{4}+\frac{3}{4}=\left(y-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Dấu ''='' xảy ra khi y = 1/2 

Vậy GTNN B là 3/4 khi y = 1/2 

c, \(C=x^2-4x+y^2-y+5=x^2-4x+4+y^2-y+\frac{1}{4}+\frac{3}{4}\)

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

Dấu ''='' xảy ra khi \(x=2;y=\frac{1}{2}\)

Vậy GTNN C là 3/4 khi x = 2 ; y = 1/2 

Bài 3 : 

a, \(x^2-6x+10=x^2-2.3.x+9+1=\left(x-3\right)^2+1\ge1>0\)( đpcm )

b, \(-y^2+4y-5=-\left(y^2-4y+5\right)=-\left(y^2-4y+4+1\right)=-\left(y-2\right)^2-1< 0\)( đpcm )

Bài 4 : 

\(B=\left(x^2+y^2\right)=\left(x+y\right)^2-2xy\)

Thay (*) ta được : \(225-2\left(-100\right)=225+200=425\)

Bài 5 : 

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

\(=2y.2x=4xy=VP\)( đpcm ) 

Trả lời:

Bài 1: 

a, \(A=x^2-4x+6=x^2-2.x.2+4+2=\left(x-2\right)^2+2\)\(\ge2\forall x\)

Dấu "=" xảy ra khi x - 2 = 0 <=> x = 2

Vậy GTNN của A = 2 khi x = 2

b, \(B=y^2-y+1=\left(y^2-2.y.\frac{1}{2}+\frac{1}{4}\right)+\frac{3}{4}=\left(y-\frac{1}{2}\right)^2+\frac{3}{4}\)\(\ge\frac{3}{4}\forall y\)

Dấu "=" xảy ra khi \(y-\frac{1}{2}=0\Leftrightarrow y=\frac{1}{2}\)

Vậy GTNN của B = 3/4 khi x = 1/2

c, \(C=x^2-4x+y^2-y+5=\left(x^2-4x\right)+\left(y^2-y\right)+4+\frac{1}{4}+\frac{3}{4}\)

\(=\left(x^2-4x+4\right)+\left(y^2-y+\frac{1}{4}\right)+\frac{3}{4}=\left(x-2\right)^2+\left(y-\frac{1}{2}\right)^2+\frac{3}{4}\)\(\ge\frac{3}{4}\forall x;y\)

Dấu "=" xảy ra khi x - 2 = 0 <=> x = 2 và y - 1/2 = 0 <=> y = 1/2

Vậy GTNN của C = 3/4 khi x = 2; y = 1/2

Bài 2: 

a, \(A=-x^2+4x+2=-\left(x^2-4x-2\right)=-\left(x^2-2.x.2+4-6\right)=-\left[\left(x-2\right)^2-6\right]\)

\(=-\left(x-2\right)^2+6\le6\forall x\)

Dấu "=" xảy ra khi x - 2 = 0 <=> x = 2

Vậy GTLN của A = 6 khi x = 2

b, \(B=x-x^2+2=-\left(x^2-x-2\right)=-\left(x^2-2.x.\frac{1}{2}+\frac{1}{4}-\frac{9}{4}\right)=-\left[\left(x-\frac{1}{2}\right)^2+\frac{9}{4}\right]\)

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

Dấu "=" xảy ra khi x - 1/2 = 0 <=> x = 1/2

Vậy GTLN của B = - 9/4 khi x = 1/2

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 \(x=\dfrac{1}{2}\)

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

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

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

Có:  \(\left(x-1\right)^2\ge0\Rightarrow\left(x-1\right)^2+4\ge4\)

Dấu '=' xảy ra khi: \(\left(x-1\right)^2=0\Rightarrow x-1=0\Rightarrow x=1\)

Vậy: \(Min_A=4\) tại \(x=1\)

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

\(B=x^2+x+\frac{1}{4}+\frac{3}{4}\)

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

Có: \(\left(x+\frac{1}{2}\right)^2\ge0\Rightarrow\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Dấu '=' xảy ra khi: \(\left(x+\frac{1}{2}\right)^2=0\Rightarrow x+\frac{1}{2}=0\Rightarrow x=-\frac{1}{2}\)

Vậy: \(Min_B=\frac{3}{4}\) tại \(x=-\frac{1}{2}\)

c) \(C=4x-x^2+3\)

\(C=-x^2+4x-4+8\) 

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

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

Có: \(\left(x-2\right)^2\ge0\Rightarrow8-\left(x-2\right)^2\le8\)

Dấu '=' xảy ra khi: \(\left(x-2\right)^2=0\Rightarrow x-2=0\Rightarrow x=2\)

Vậy: \(Max_C=8\) tại \(x=2\)

a. \(P=x^2-2x+5=x^2-2x+1+4=\left(x-1\right)^2+4\)

vì \(\left(x-1\right)^2\ge0\) với mọi x

=> (x-1)^2 +4 \(\ge\) vợi mọi x

Pmin=4 <=> x-1=0 <=>x=1

1.

b)\(M=\left(x^2-x+\frac{1}{4}\right)+\left(y^2+6y+9\right)+\frac{3}{4}=\left(x-\frac{1}{2}\right)^2+\left(y+3\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Dấu = xảy ra \(\Leftrightarrow x-\frac{1}{2}=0\) và \(y+3=0\)

\(\Leftrightarrow x=\frac{1}{2}\) và \(y=-3\)

Vậy GTNN của M là \(\frac{3}{4}\Leftrightarrow x=\frac{1}{2}\)và \(y=-3\)

1.

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

Dấu = xảy ra \(\Leftrightarrow x-1=0\Leftrightarrow x=1\)

Vậy GTNN của P là 4