a)a²+b²+c²+3=2(a+b+c)

=>a²+b²+c²+1+1+1-2a-2b-2c=0

=>(a²-2a+1)+(b²-2b+1)+(c²-2c+1)=0

=>(a-1)²+(b-1)²+(c-1)²=0

=>a-1=b-1=c-1=0 <=>a=b=c=1 

-->Đpcm

b)(a+b+c)²=3(ab+ac+bc)

=>a²+b²+c²+2ab+2ac+2bc -3ab-3ac-3bc=0 

=>a²+b²+c²-ab-ac-bc=0

=>2a²+2b²+2c²-2ab-2ac-2bc=0 

=>(a²- 2ab+b²)+(b²-2bc+c²) + (c²-2ca+a²) = 0

=>(a-b)²+(b-c)²+(c-a)²=0 

Hay (a-b)²=0 hoặc (b-c)²=0 hoặc (a-c)²=0

=>a-b hoặc b=c hoặc a=c

=>a=b=c 

-->Đpcm

c)a²+b²+c²=ab+bc+ca

=>2(a²+b²+c²)=2(ab+bc+ca)

=>2a²+2b²+c²=2ab+2bc+2ca

=>2a²+2b²+c²-2ab-2bc-2ca=0

=>a²+a²+b²+b²+c²+c²-2ab-2bc-2ca=0

=>(a²-2ab+b²)+(b²-2bc+c²)+(a²-2ca+c²)=0

=>(a-b)²+(b-c)²+(a-c)²=0

Hay (a-b)²=0 hoặc (b-c)²=0 hoặc (a-c)²=0

=>a-b hoặc b=c hoặc a=c

=>a=b=c 

-->Đpcm

a) Ta có : \(a^2+b^2+c^2+3=2\left(a+b+c\right)\)

\(\Leftrightarrow\left(a^2-2a+1\right)+\left(b^2-2b+1\right)+\left(c^2-2c+1\right)=0\)

\(\Leftrightarrow\left(a-1\right)^2+\left(b-1\right)^2+\left(c-1\right)^2=0\)

Vì \(\left(a-1\right)^2\ge0,\left(b-1\right)^2\ge0,\left(c-1\right)^2\ge0\) nên pt trên tương đương với \(\begin{cases}\left(a-1\right)^2=0\\\left(b-1\right)^2=0\\\left(c-1\right)^2=0\end{cases}\) \(\Leftrightarrow a=b=c=1\)

b) \(\left(a+b+c\right)^2=3\left(ab+bc+ac\right)\)

\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ac\right)=3\left(ab+bc+ac\right)\)

\(\Leftrightarrow a^2+b^2+c^2=ab+bc+ac\) (1)

\(\Leftrightarrow2\left(a^2+b^2+c^2\right)=2ab+2bc+2ac\)

\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ac+a^2\right)=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

Mà \(\left(a-b\right)^2\ge0,\left(b-c\right)^2\ge0,\left(c-a\right)^2\ge0\)

\(\Rightarrow\begin{cases}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{cases}\) \(\Rightarrow a=b=c\)

c) Giải tương tự câu b) , bắt đầu từ (1)

b, ta có a³+ b³ = (a+b)(a²-ab +b²)

= (a+b)(a² -ab +b² -ab +ab)

= (a+b) ( a²-2ab +b +ab)

=(a+b) [ (a²-b²) +ab ]

vậy ...........................

câu a bạn sai đề à

đề c/ m : ( a+b)(a²-ab +b²) +(a-b)(a²+ab+b² ) =2a³

(a+ b)(a² -ab +b²) + (a-b)(a²+ab +b²)

= a³+b³+a³-b³(hdt)

= 2a³

^{chúc bạn học tốt}

VT = ( a + b )(a^2 - ab + b^2) + ( a-  b)(a^2 + ab + b^2) 

    = a^3 + b^3 + a^3 - b^3

     = 2a^3 

    =VP

=> ĐPCM 

a)\(a^2+ab+b^2=a^2+\dfrac{2ab}{2}+\left(\dfrac{b}{2}\right)^2+\dfrac{3b^2}{4}\)

\(=\left(a+\dfrac{b}{2}\right)^2+\dfrac{3b^2}{4}\ge0\forall a,b\)

b)\(a^4+b^4\ge a^3b+ab^3\)

\(\Leftrightarrow a^3\left(a-b\right)-b^3\left(a-b\right)\ge0\)

\(\Leftrightarrow\left(a^3-b^3\right)\left(a-b\right)\ge0\)

\(\Leftrightarrow\left(a-b\right)^2\left(a^2+ab+b^2\right)\ge0\forall a,b\)

a³-b³+ab(a-b)

=(a-b)(a²+ab+b²)+ab(a-b) (HĐT số 7)

=(a-b)(a²+ab+ab+b²)

=(a-b)(a²+2ab+b²)

=(a-b)(a+b)²  (HĐT số 1)

Đpcm

a. \(2\left(a^2+b^2\right)=\left(a-b\right)^2\)

\(\Leftrightarrow2a^2+2b^2=a^2+b^2-2ab\)

\(\Leftrightarrow a^2+b^2=-2ab\)

\(\Leftrightarrow a^2+2ab+b^2=0\)

\(\Leftrightarrow\left(a+b\right)^2=0\)

\(\Leftrightarrow a+b=0\Leftrightarrow a=-b\) (đpcm)

b. \(a^2+b^2+c^2+3=2\left(a+b+c\right)\)

\(\Leftrightarrow a^2+b^2+c^2+3-2a-2b-2c=0\)

\(\Leftrightarrow\left(a^2-2a+1\right)+\left(b^2-2b+1\right)+\left(c^2-2c+1\right)=0\)

\(\Leftrightarrow\left(a-1\right)^2+\left(b-1\right)^2+\left(c-1\right)^2=0\)

Vì \(\left(a-1\right)^2;\left(b-1\right)^2;\left(c-1\right)^2\ge0\)

\(\Rightarrow\left(a-1\right)^2=\left(b-1\right)^2=\left(c-1\right)^2=0\)

\(\Leftrightarrow a-1=b-1=c-1=0\Leftrightarrow a=b=c=1\)

c. \(\left(a+b+c\right)^2=3\left(ab+bc+ca\right)\)

\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=3\left(ab+bc+ca\right)\)

\(\Leftrightarrow a^2+b^2+c^2=ab+bc+ca\)

\(\Leftrightarrow2\left(a^2+b^2+c^2\right)=2\left(ab+bc+ca\right)\)

\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

Tương tự câu b ta có a = b = c

1) \(\Sigma\frac{a}{b^3+ab}=\Sigma\left(\frac{1}{b}-\frac{b}{a+b^2}\right)\ge\Sigma\frac{1}{a}-\Sigma\frac{1}{2\sqrt{a}}=\Sigma\left(\frac{1}{a}-\frac{2}{\sqrt{a}}+1\right)+\Sigma\frac{3}{2\sqrt{a}}-3\)

\(\ge\Sigma\left(\frac{1}{\sqrt{a}}-1\right)^2+\frac{27}{2\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)}-3\ge\frac{27}{2\sqrt{3\left(a+b+c\right)}}-3=\frac{3}{2}\)

2.

Vỉ \(ab+bc+ca+abc=4\)thi luon ton tai \(a=\frac{2x}{y+z};b=\frac{2y}{z+x};c=\frac{2z}{x+y}\)

\(\Rightarrow VT=2\Sigma_{cyc}\sqrt{\frac{ab}{\left(b+c\right)\left(c+a\right)}}\le2\Sigma_{cyc}\frac{\frac{b}{b+c}+\frac{a}{c+a}}{2}=3\)

Cho o dong 2 la x,y,z nhe,ghi nham

a) Ta có:

(a + b)² >= 0 => a² + b² >= -2ab

(a - 1)² >= 0 => a² + 1 >= 2a

(b - 1)² >= 0 => b² + 1 >= 2b

Cộng từng vế ta được: 2a² +2b² +2 >= -2ab + 2a +2b => a² + b² + 1 >= -ab + a + b

Dấu "=" xảy ra khi a= - b; a = 1; b = 1 không đạt được nên không xảy ra dấu bằng do đó:

a² + b² + 1 > -ab + a + b      .đpcm.

b) a + b + c = 0 => a + b = -c => (a + b)³ = -c³  => a³ + 3a²b +3 ab² + b³ = -c³

=> a³ + b³ + c³ = -3ab(a + b)   (*)

Mà a + b + c = 0 => a + b = -c 

=> (*) <=>  a³ + b³ + c³ = 3abc     .đpcm.