\(\dfrac{2}{a+2}+\dfrac{2}{b+2}+\dfrac{2}{c+2}\ge2\)

\(\Leftrightarrow\dfrac{2}{a+2}-1+\dfrac{2}{b+2}-1+\dfrac{2}{c+2}-1\ge2-3\)

\(\Rightarrow1\ge\dfrac{a}{a+2}+\dfrac{b}{b+2}+\dfrac{c}{c+2}=\dfrac{a^2}{a^2+2a}+\dfrac{b^2}{b^2+2b}+\dfrac{c^2}{c^2+2c}\)

\(\Rightarrow1\ge\dfrac{\left(a+b+c\right)^2}{a^2+2a+b^2+2b+c^2+2c}\)

\(\Rightarrow a^2+b^2+c^2+2\left(a+b+c\right)\ge a^2+b^2+c^2+2\left(ab+bc+ca\right)\)

\(\Rightarrow\) đpcm

Phía trên thoả mãn \(\ge1\) chứ không phải 3/2 đâu ạ 

Áp dụng BĐT Cosi:

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

Cmtt: \(\dfrac{b}{\sqrt{c^2+bc}}\ge\dfrac{2\sqrt{2}b}{b+3c};\dfrac{c}{\sqrt{a^2+ca}}\ge\dfrac{2\sqrt{2}c}{c+3a}\)

\(\Leftrightarrow P\ge2\sqrt{2}\left(\dfrac{a}{a+3b}+\dfrac{b}{b+3c}+\dfrac{c}{c+3a}\right)\\ \Leftrightarrow\dfrac{P}{\sqrt{2}}\ge2\left(\dfrac{a}{a+3b}+\dfrac{b}{b+3c}+\dfrac{c}{c+3a}\right)\\ \Leftrightarrow\dfrac{P}{\sqrt{2}}\ge\dfrac{2\left(a+b+c\right)^2}{a^2+b^2+c^2+3\left(ab+bc+ca\right)}\ge\dfrac{2\left(a+b+c\right)^2}{\left(a+b+c\right)^2+\dfrac{1}{3}\left(a+b+c\right)^2}\\ \Leftrightarrow\dfrac{P}{\sqrt{2}}\ge\dfrac{2}{\dfrac{4}{3}}=\dfrac{3}{2}\\ \Leftrightarrow P\ge\dfrac{3\sqrt{2}}{2}\)

Dấu \("="\Leftrightarrow a=b=c\)

từ dòng thứ 4 lm sao suy ra dòng thứ 5 thế ạ

Vế trái bậc 0, vế phải bậc 1, không đồng bậc với nhau . BĐT sai ngay với \(a=9,b=3,c=6\)

Sửa: \(\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}\geq \frac{3(a^2+b^2+c^2)}{ab+bc+ac}\)

Chứng minh:

Áp dụng BĐT Cauchy-Schwarz:

\(\text{VT}=\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=\frac{a^4}{a^2bc}+\frac{b^4}{b^2ac}+\frac{c^4}{c^2ab}\)

\(\geq \frac{(a^2+b^2+c^2)^2}{a^2bc+b^2ac+c^2ab}=\frac{(a^2+b^2+c^2)^2}{abc(a+b+c)}(1)\)

Ta có kết quả quen thuộc của BĐT Cauchy là:

\(a^2+b^2+c^2\geq ab+bc+ac\)

Và: \((ab+bc+ac)^2\geq 3abc(a+b+c)\)

Do đó: \(a^2+b^2+c^2\geq ab+bc+ac\geq \frac{3abc(a+b+c)}{ab+bc+ac}(2)\)

Từ \((1);(2)\Rightarrow \text{VT}\geq \frac{(a^2+b^2+c^2).3abc(a+b+c)}{(ab+bc+ac)abc(a+b+c)}=\frac{3(a^2+b^2+c^2)}{ab+bc+ac}\) (đpcm)

Dấu bằng xảy ra khi $a=b=c$

xin lỗi là \(a^3+b^3+c^3\) nhé

Đặt: \(L=\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ac+a^2}\)

Áp dụng bất đẳng thức AM-GM:

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

Chứng minh tương tự: \(\left\{{}\begin{matrix}\dfrac{b^3}{b^2+bc+c^2}\ge\dfrac{b^3}{\dfrac{3}{2}\left(b^2+c^2\right)}\\\dfrac{c^3}{c^2+ac+a^2}\ge\dfrac{c^3}{\dfrac{3}{2}\left(c^2+a^2\right)}\end{matrix}\right.\)

Cộng theo vế: \(L\ge\dfrac{2}{3}\left(\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2}\right)\)

Tiếp tục áp dụng AM-GM:

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

Chứng minh tương tự: \(\left\{{}\begin{matrix}\dfrac{b^3}{b^2+c^2}\ge b-\dfrac{c}{2}\\\dfrac{c^3}{c^2+a^2}\ge c-\dfrac{a}{2}\end{matrix}\right.\)

Cộng theo vế:

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

Đặt \(\left(a;b;c\right)=\left(\dfrac{y}{x};\dfrac{z}{y};\dfrac{x}{z}\right)\)

\(\Rightarrow VT=\dfrac{1}{\dfrac{y}{x}\left(\dfrac{z}{y}+1\right)}+\dfrac{1}{\dfrac{z}{y}\left(\dfrac{x}{z}+1\right)}+\dfrac{1}{\dfrac{x}{z}\left(\dfrac{y}{x}+1\right)}\)

\(VT=\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=\dfrac{x^2}{xy+xz}+\dfrac{y^2}{xy+yz}+\dfrac{z^2}{xz+yz}\)

\(VT\ge\dfrac{\left(x+y+z\right)^2}{2\left(xy+yz+zx\right)}\ge\dfrac{3\left(xy+yz+zx\right)}{2\left(xy+yz+zx\right)}=\dfrac{3}{2}\)

Áp dụng bất đẳng thức Cô si cho hai số dương ta có:

(a2 + b2) + (b2 + c2) + (c2 + a2) ≥ 2ab + 2bc + 2ca

=> 2(a2 + b2 + c2 ) ≥ 2 (ab + bc + ca) (1) (a2 + 1) + (b2 + c2) + (c2 + a2) ≥ 2a + 2b + 2c

=> a2 + b2 + c2 + 3 ≥ 2(a + b + c) (2)

Cộng các vế của (1) và (2) ta có:

3 ( a2 + b2 + c2 ) + 3 ≥ 2 (ab + bc + ca + a + b + c)

=> 3( a2 + b2 + c2 ) + 3 ≥ 12 => a2 + b2 + c2 ≥ 3.

Ta có: (a^3/b + ab ) + ( b^3/c + bc ) + ( c^3/a + ca)≥ 2(a2 + b2 + c2) (CÔ SI) 

<=>a^3/b + b^3/c + c^3/a +ab + bc + ac  ≥ 2(a2 + b2 + c2)

Vì a2 + b2 + c2 ≥ ab + bc + ca => a^3 + b^3 + c^3 ≥ a2 + b2 + c2 ≥ 3 (đpcm).

Áp dụng bất đẳng thức cô-si cho hai số dương ta có:

\(\left(a^2+b^2\right)+\left(b^2+c^2\right)+\left(c^2+a^2\right)\ge2ab+2bc+2ca\)

\(\Rightarrow2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ca\right)\) (1)

\(\left(a^2+b^2\right)+\left(b^2+c^2\right)+\left(c^2+a^2\right)\ge2a+2b+2c\)

\(\Rightarrow a^2+b^2+c^2+3\ge2\left(a+b+c\right)\) (2)

Cộng (1) với (2)

\(3\left(a^2+b^2+c^2\right)+3\ge2\left(ab+bc+ca+a+b+c\right)\)

\(\Rightarrow3\left(a^2+b^2+c^2\right)+3\ge12\)

\(\Rightarrow a^2+b^2+c^2\ge3\)

Ta có: \(\left(\dfrac{a^3}{b}+ab\right)+\left(\dfrac{b^3}{c}+bc\right)+\left(\dfrac{c^3}{a}+ca\right)\ge2\left(a^2+b^2+c^2\right)\)

\(\Leftrightarrow\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a}+ab+bc+ca\ge2\left(a^2+b^2+c^2\right)\)

Vì \(a^2+b^2+c^2\ge ab+bc+ca\)

\(\Rightarrow\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a}\ge a^2+b^2+c^2\ge3\) (đpcm).