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

\(\Leftrightarrow a^2+b^2+c^2-ab-bc-ca=0\) (chuyển vế qua)

\(\Leftrightarrow\frac{1}{2}\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]=0\)

Do VP >=0 với mọi a, b, c. Nên để đăng thức xảy ra thì a = b = c

c) a + b + c = 0 suy ra a = -(b+c)

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

\(=b^3+c^3-b^3-3bc\left(b+c\right)-c^3\)

\(=3bc.\left[-\left(b+c\right)\right]=3abc\) (đpcm)

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

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

Do VT >=0 với mọi a, b, c nên a = b = c 1

tí đăng tiếp

a) Xét vế trái: \(\left(a+b+c\right)^3-a^3-b^3-c^3\)

\(=\left(a+b\right)^3+3a^2bc+3abc^2+c^3-a^3-b^3-c^3\)

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

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

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

\(=3\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]\)

\(=3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

b) \(a^3+b^3+c^3-3abc\)

\(=\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc\)

\(=\left(a+b+c\right)\left[\left(a+b\right)^2-c\left(a+b\right)+c^2\right]-3ab\left(a+b+c\right)\)

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

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

Chúc bạn học tốt và nhớ k cho mình với nhá!

ngu

`(a+b+c)^2=3(ab+bc+ca)`

`<=>a^2+b^2+c^2+2ab+2bc+2ca=3(ab+bc+ca)`

`<=>a^2+b^2+c^2=ab+bc+ca`

`<=>2a^2+2b^2+2c^2=2ab+2bc+2ca`

`<=>(a-b)^2+(b-c)^2+(c-a)^2=0`

`VT>=0`

Dấu "=" xảy ra khi `a=b=c`

`a^3+b^3+c^3=3abc`

`<=>a^3+b^3+c^3-3abc=0`

`<=>(a+b)^3+c^3-3abc-3ab(a+b)=0`

`<=>(a+b)^3+c^3-3ab(a+b+c)=0`

`<=>(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0`

`**a+b+c=0`

`**a^2+b^2+c^2=ab+bc+ca`

`<=>a=b=c`

Áp dụng BĐT AG-GM:

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

Cmtt \(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{b^3}{b^2+bc+c^2}\ge\dfrac{b^3}{\dfrac{3}{2}\left(b^2+c^2\right)}\\\dfrac{c^3}{c^2+ac+a^2}\ge\dfrac{c^3}{\dfrac{3}{2}\left(c^2+a^2\right)}\end{matrix}\right.\)

Cộng vế theo vế của bất đẳng thức:

\(\Leftrightarrow VT\ge\dfrac{2}{3}\left(\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2}\right)\)

Tiếp tục áp dụng BĐT AG-GM:

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

Cmtt\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{b^3}{b^2+c^2}\ge b-\dfrac{c}{2}\\\dfrac{c^3}{c^2+a^2}\ge c-\dfrac{a}{2}\end{matrix}\right.\)

Cộng vế theo vế

\(\Leftrightarrow VT\ge\dfrac{2}{3}\left(\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2}\right)\\ \ge\dfrac{2}{3}\left(a-\dfrac{b}{2}+b-\dfrac{c}{2}+c-\dfrac{a}{2}\right)=\dfrac{2}{3}\left(a+b+c-\dfrac{a+b+c}{2}\right)=\dfrac{a+b+c}{3}\)

\(\dfrac{a^3}{a^2+ab+b^2}=a-\dfrac{ab\left(a+b\right)}{a^2+ab+b^2}\ge a-\dfrac{ab\left(a+b\right)}{3\sqrt[3]{a^2.ab.b^2}}=a-\dfrac{a+b}{3}=\dfrac{2a-b}{3}\)

Tương tự và cộng lại ta sẽ có đpcm