Ta có : x² + 2x + 2

= x² + 2x + 1 + 1

= (x + 1)² + 1 \(\ge1\forall x\)

Vậy  x² + 2x + 2 \(>0\forall x\)

Ta có : x² + 2x + 2

=> x² + 2x + 1 + 1

=> ( x + 1)² + 1  >  1\(\forall x\)

Vậy x² + 2x + 2   > \(0\forall x\)

a) \(-\left(x^2-6x+10\right)=-\left(x^2-6x+9+1\right)=-\left[\left(x-3\right)^2+1\right]\le-1< 0\forall x\)

BĐT đúng

b) \(x^2+x+1=x^2+2.x.\frac{1}{2}+\frac{1}{4}+\frac{3}{4}=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}>0\forall x\)

BĐT đúng

c)Dấu "=" ko xảy ra???

\(=\left(4x^2+2.2x.y+y^2\right)+2\left(2x+y\right)+1+2\)

\(=\left(2x+y\right)^2+2.\left(2x+y\right).1+1+1\)

\(=\left(2x+y+1\right)^2+1\ge1>0\) (đpcm)

a. −x² + 6x - 10

= x² − 6x) − 10

= x² − 2.x.3 + 3² − 9) − 10

= x − 3)² + 9 − 10

= x − 3)² −1

Vì x − 3)² ≥ 0 ∀ x ⇒ x − 3)² ≤ 0 ⇒ x − 3)² −1 ≤ −1

Vậy x − 3)² −1 < 0 ⇒ −x² + 6x - 10 luôn âm với mọi x

b. x² + x + 1

= x² + 2.x.\(\frac{1}{2}\)+ (\(\frac{1}{2}\))²

)² + \(\frac{3}{4}\)

Vì (x + \(\frac{1}{2}\))² ≥ 0 ∀ x ⇒ (x + \(\frac{1}{2}\))² + \(\frac{3}{4}\) ≥ \(\frac{3}{4}\) ∀ x

Vậy (x + \(\frac{1}{2}\))² + \(\frac{3}{4}\) ≥ 0 hay x² + x + 1 > 0 ∀ x.

5x^2+2y^2+4xy-4x-y+5=(4x^2+y^2+4xy)+(x^2-4x+4)+(y^2-y+1/4)+3/4 =(2x+y)^2+(x-2)^2+(y-1/2)^2+3/4  (1)

 vi (2x+y)^2>=0 , (x-2)^2>=0  ,(y-1/2)^2>=0 (2)

tu 1 va 2 suy ra dieu phai chung minh

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

a) Xét \(x^2-4x+4=\left(x-2\right)^2\ge0\)

<=> \(x^2-4x\ge-4>-5\)

b) \(2x^2+4y^2-4x-4xy+5\)

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

= \(\left(x-2\right)^2+\left(x-2y\right)^2+1\ge1>0\)

a)

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

\(=\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}+1\ge0+0+1=1\)

mà\(1>0\Rightarrow x^2+xy+y^2+1>0\)với mọi \(x\)và\(y\)

b)

\(x^2+5y^2+2x-4xy-10y+14\)

\(=\left[x^2+2x\left(1-2y\right)+\left(1-2y\right)^2\right]+y^2-6y+13\)

\(=\left(x+1-2y\right)^2+\left(y^2-2y\times3+9\right)+4\)

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

Ta có:\(\left(x+1-2y\right)^2\ge0\)với mọi \(x;y\in R\)

và\(\left(y-3\right)^2\ge0\)với mọi \(x;y\in R\)

\(\Rightarrow\left(x+1-2y\right)^2+\left(y-3\right)^2+4\ge4\)với mọi \(x;y\in R\)

\(\Rightarrow x^2+5y^2+2x-4xy-10y+14>0\)

c)

\(5x^2+10y^2-6xy-4x-2y+3=x^2+4x^2+y^2+9y^2-6xy-4x-2y+3\)

\(=\left[\left(2x\right)^2-2\times2x+1\right]+\left(y^2-2y+1\right)+\left[\left(3y\right)^2-2\times3y+x^2\right]+1\)

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

Ta có \(\left(2x+1\right)^2\ge0\)với mọi  \(x\)

\(\left(y-1\right)^2\ge\)với mọi \(y\)

\(\left(3y-x\right)^2\ge0\)với mọi \(x;y\)

và \(1>0\)

\(\Rightarrow5x^2+10y^2-6xy-4x-2y+3>0\)

a. \(x^2+xy+y^2+1=\left(x^2+xy+\frac{1}{4}y^2\right)+\frac{3}{4}y^2+1=\left(x+\frac{1}{4}y\right)^2+\frac{3}{4}y^2+1>0\forall x;y\)(đpcm)

b. \(x^2+5y^2+2x-4xy-10y+14\)

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

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

\(=\left(x-2y-1\right)^2+\left(y-3\right)^2+4>0\forall x;y\)(đpcm)

c.  tương tự ý b

_______________Bài làm___________________

a, \(x^2+xy+y^2+1\)

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

Do \(\left(x+\dfrac{y}{2}\right)^2\ge0\forall x,y\)

Và \(\dfrac{3y^2}{4}\ge0\forall y\)

Nên: \(\left(x+\dfrac{y}{2}\right)^2+\dfrac{3y^2}{4}+1>0\forall x,y=>đpcm\)

b, \(x^2+5y^2+2x-4xy-10y+14\)

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

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

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

Do \(\left(x-2y+1\right)^2\ge0\forall x,y\)

Và \(\left(y-3\right)^2\ge0\forall y\)

Nên \(\left(x-2y+1\right)^2+\left(y-3\right)^2+4>0\)

c, \(5x^2+10y^2-6xy-4x-2y+3\)

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

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

Do .........

tự làm ik

a, \(x^2+xy+y^2+1=x^2+\dfrac{1}{2}xy+\dfrac{1}{2}xy+\dfrac{1}{4}y^2+\dfrac{3}{4}y^2+1\)

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

Với mọi giá trị của \(x;y\in R\) ta có:

\(\left(x^2+\dfrac{1}{2}y\right)^2+\dfrac{3}{4}y^2\ge0\)

\(\Rightarrow\left(x^2+\dfrac{1}{2}y\right)^2+\dfrac{3}{4}y^2+1\ge1\)

Vậy............

b, \(5x^2+10y^2-6xy-4x-2y+3\)

\(=x^2-6xy+9y^2+4x^2-4x+1+y^2-2y+1+1\)

\(=x^2-3xy-3xy+9y^2+4x^2-2x-2x+1+y^2-y-y+1+1\)

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

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

Với mọi giá trị của \(x;y\in R\) ta có:

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

\(\Rightarrow\left(x-3y\right)^2+\left(2x-1\right)^2+\left(y-1\right)^2+1\ge1\)

Vậy..............

Chúc bạn học tốt!!!