do \(a,b,c\ge1\)\(=>\left\{{}\begin{matrix}b+c\ge2\\c+a\ge2\\a+b\ge2\end{matrix}\right.\)

\(=>\left\{{}\begin{matrix}a\left(b+c\right)\ge2a\\b\left(c+a\right)\ge2b\\c\left(a+b\right)\ge2c\end{matrix}\right.\)

\(=>\) biểu thức đề bài cho\(\ge2\left(a+b+c+\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\)

\(2\left(1+1+1+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}\right)=9\)

dấu= xảy ra<=>a=b=c=1

\(a;b;c\ge1\Rightarrow\left\{{}\begin{matrix}ab;bc;ca\ge1\\ab+bc+ca\ge3\end{matrix}\right.\)

Ta có:

\(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}=\dfrac{a^2+b^2+2}{a^2b^2+a^2+b^2+1}=1-\dfrac{a^2b^2-1}{a^2b^2+a^2+b^2+1}\ge1-\dfrac{a^2b^2-1}{a^2b^2+2ab+1}=\dfrac{2}{ab+1}\)

Tương tự: \(\dfrac{1}{a^2+1}+\dfrac{1}{c^2+1}\ge\dfrac{2}{ac+1}\) ; \(\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\ge\dfrac{2}{bc+1}\)

Cộng vế: \(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\ge\dfrac{1}{ab+1}+\dfrac{1}{bc+1}+\dfrac{1}{ca+1}\)

Do đó: \(VT\ge2\left(ab+bc+ca+\dfrac{1}{ab+1}+\dfrac{1}{bc+1}+\dfrac{1}{ca+1}\right)\)

\(VT\ge2\left(ab+bc+ca+\dfrac{9}{ab+bc+ca+3}\right)\)

Đặt \(ab+bc+ca=x\ge3\Rightarrow VT\ge2\left(x+\dfrac{9}{x+3}\right)\)

\(VT\ge2\left(\dfrac{x+3}{4}+\dfrac{9}{x+3}+\dfrac{3x}{4}-\dfrac{3}{4}\right)\ge2\left(2\sqrt{\dfrac{9\left(x+3\right)}{4\left(x+3\right)}}+\dfrac{3}{4}.3-\dfrac{3}{4}\right)=9\)

Bài 1: 

a) Áp dụng bđt Cô - si:

\(\dfrac{a}{b^2}+\dfrac{1}{a}\ge\dfrac{2}{b}\)

Tương tự với 2 phân thức còn lại của vế trái rồi cộng lại, ta có:

\(\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{2}{a}+\dfrac{2}{b}+\dfrac{2}{c}\)

=> đpcm

Bài dù a + b + c = 2021 hay 1 số bất kì thì bđt luôn \(\ge\dfrac{3}{2}\). Bạn có thể tham khảo bđt Nesbitt

Bài 2:

\(P=\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)

\(=\dfrac{2021-\left(b+c\right)}{b+c}+\dfrac{2021-\left(c+a\right)}{c+a}+\dfrac{2021-\left(a+b\right)}{a+b}\)

\(=2021\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)-3\)

Áp dụng BĐT Svacxo, ta có

\(P\) ≥ \(\dfrac{9}{2}-3=\dfrac{3}{2}\)

Dấu"=" ⇔ ...

Sau khi đã đi tham khảo 7749 người thì đã cho ra một kết quả:v

Bài 2. \(P=\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)

\(P=\dfrac{a}{b+c}+1+\dfrac{b}{c+a}+1+\dfrac{c}{a+b}+1-3\)

\(P=\dfrac{a+b+c}{b+c}+\dfrac{a+b+c}{c+a}+\dfrac{a+b+c}{a+b}-3\)

\(P=\dfrac{(2a+2b+3c)( \dfrac{1}{b+c}+\dfrac{1}{c+a}+\dfrac{1}{a+b})}{2}-3 ≥ \dfrac{9}{2}-3=\dfrac{3}{2}\)

Dấu `"="` xảy ra:

\(\Leftrightarrow \begin{cases} a=b=c\\ a+b+c=2021 \end{cases} \)

\(\Leftrightarrow a=b=c=\dfrac{2021}{3}\)

Vậy \(min \) \(P=\dfrac{3}{2}\) khi \(a=b=c=\dfrac{2021}{3}\)

Đây là BĐT Iran 96 khá nổi tiếng. Bạn hoàn toàn có thể search trên google lời giải.

\(a+\dfrac{1}{a+1}=\dfrac{a^2+a+1}{a+1}=\dfrac{4a^2+4a+4}{4\left(a+1\right)}=\dfrac{3\left(a+1\right)^2+\left(a-1\right)^2}{4\left(a+1\right)}\ge\dfrac{3\left(a+1\right)^2}{4\left(a+1\right)}=\dfrac{3}{4}\left(a+1\right)\ge\dfrac{3}{2}\sqrt{a}\)

Tương tự: \(b+\dfrac{1}{b+1}\ge\dfrac{3}{2}\sqrt{b}\) ; \(c+\dfrac{1}{c+1}\ge\dfrac{3}{2}\sqrt{c}\)

Nhân vế:

\(VT\ge\dfrac{27}{8}\sqrt{abc}\ge\dfrac{27}{8}\) (đpcm)

\(\Leftrightarrow\left(1+abc\right)\left(\dfrac{1}{a\left(1+b\right)}+\dfrac{1}{b\left(1+c\right)}+\dfrac{1}{c\left(1+a\right)}\right)\ge3\)

Ta có:

\(\left(1+abc\right).\dfrac{1}{a\left(1+b\right)}=\dfrac{1+abc}{a+ab}=\dfrac{1+a+ab+abc-a-ab}{a+ab}=\dfrac{1+a}{a\left(1+b\right)}+\dfrac{b\left(1+c\right)}{1+b}-1\)

\(\Rightarrow VT=\dfrac{1+a}{a\left(1+b\right)}+\dfrac{b\left(1+c\right)}{1+b}+\dfrac{1+b}{b\left(1+c\right)}+\dfrac{c\left(1+a\right)}{1+c}+\dfrac{1+c}{c\left(1+a\right)}+\dfrac{a\left(1+b\right)}{1+a}-3\)

\(VT\ge6\sqrt[6]{\dfrac{abc\left(1+a\right)^2\left(1+b\right)^2\left(1+c\right)^2}{abc\left(1+a\right)^2\left(1+b\right)^2\left(1+c\right)^2}}-3=3\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=1\)

Chính bài của em:

Cho \(a,b,c\ge1\). CMR: \(a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}... - Hoc24

Lời giải:

Ta có:

\(\text{VT}=\frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)}\)

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

\(=\frac{ab+bc+ac+a+b+c}{2+(a+b+c)+ab+bc+ac}\)

Ta cần chứng minh \(\text{VT}\geq \frac{3}{4}\)

\(\Leftrightarrow \frac{ab+bc+ac+a+b+c}{2+(a+b+c)+ab+bc+ac}\geq \frac{3}{4}\)

\(\Leftrightarrow 4(ab+bc+ac+a+b+c)\geq 3(ab+bc+ac+a+b+c)+6\)

\(\Leftrightarrow ab+bc+ac+a+b+c\geq 6\)

\(\Leftrightarrow ab+bc+ac+a+b+c\geq 6\sqrt[6]{ab.bc.ac.a.b.c}\)

(Đúng theo BĐT Cô-si)

Do đó ta có đpcm

Dấu bằng xảy ra khi \(a=b=c=1\)

Giải:

\(\dfrac{a}{\left(a+1\right)\left(b+1\right)}+\dfrac{b}{\left(b+1\right)\left(c+1\right)}+\dfrac{c}{\left(c+1\right)\left(a+1\right)}\ge\dfrac{3}{4}\)(*)

\(\Leftrightarrow\) \(\dfrac{a\left(c+1\right)+b\left(a+1\right)+c\left(b+1\right)}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}\ge\dfrac{3}{4}\)

\(\Leftrightarrow\) \(\dfrac{ac+a+ab+b+bc+c}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}\) \(\ge\) \(\dfrac{3}{4}\)

Do a+1 ; b+1; c+1 >0

\(\Rightarrow\) 4ac+4a+4ab+4b+4bc+4c \(\ge\) 3abc+3ac+3bc+3ab+3a+3b+3c+3

\(\Leftrightarrow\) ac+ab+bc+a+b+c -6 \(\ge\) 0

Áp dụng BĐT Cô-si cho 3 số

Ta có: a+b+c \(\ge\) \(3\sqrt[3]{abc}=3\)

ab+bc+ca \(\ge\) \(3\sqrt[3]{\left(abc\right)^2}\) = 3

\(\Rightarrow\)ac+ab+bc+a+b+c -6 \(\ge\) 0 ( luôn đúng)

\(\Rightarrow\) (*) được chứng minh

Dấu "=" xảy ra \(\Leftrightarrow\) a=b=c=1

wow

wow, chắc xu học lớp 9

\(\dfrac{a^3}{\left(a+b\right)\left(b+c\right)}+\dfrac{a+b}{8}+\dfrac{b+c}{8}\ge3\sqrt[3]{\dfrac{a^3\left(a+b\right)\left(b+c\right)}{64}}=\dfrac{3a}{4}\)

Tương tự:

\(\dfrac{b^3}{\left(b+c\right)\left(c+a\right)}+\dfrac{b+c}{8}+\dfrac{c+a}{8}\ge\dfrac{3b}{4}\)

\(\dfrac{c^3}{\left(c+a\right)\left(a+b\right)}+\dfrac{c+a}{8}+\dfrac{a+b}{8}\ge\dfrac{3c}{4}\)

Cộng vế:

\(VT+\dfrac{4\left(a+b+c\right)}{8}\ge\dfrac{3\left(a+b+c\right)}{4}\)

\(\Rightarrow VT\ge\dfrac{a+b+c}{4}\)

Dấu "=" xảy ra khi \(a=b=c\)