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Áp dụng BĐT AM-GM ta có: 

\(\frac{1}{d+1}=1-\frac{d}{d+1}\ge\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\)

\(\ge3\sqrt[3]{\frac{abc}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}}\). TƯơng tự cho 3 BĐT còn lại

\(\frac{1}{a+1}\ge3\sqrt[3]{\frac{bcd}{\left(b+1\right)\left(c+1\right)\left(d+1\right)}};\frac{1}{b+1}\ge3\sqrt[3]{\frac{acd}{\left(a+1\right)\left(c+1\right)\left(d+1\right)}};\frac{1}{c+1}\ge3\sqrt[3]{\frac{abd}{\left(a+1\right)\left(b+1\right)\left(d+1\right)}}\)

Nhân theo vế 4 BDT trên ta có: 

\(\frac{1}{\left(a+1\right)\left(b+1\right)\left(c+1\right)\left(d+1\right)}\ge81\sqrt[3]{\left(\frac{abcd}{\left(a+1\right)\left(b+1\right)\left(c+1\right)\left(d+1\right)}\right)^3}\)

\(\Leftrightarrow\frac{1}{\left(a+1\right)\left(b+1\right)\left(c+1\right)\left(d+1\right)}\ge\frac{81abcd}{\left(a+1\right)\left(b+1\right)\left(c+1\right)\left(d+1\right)}\)

Hay ta có ĐPCM

Cosi + Svac-xơ

Có : \(3=a^2+b^2+c^2\ge\frac{\left(a+b+c\right)^2}{3}\)\(\Leftrightarrow\)\(a+b+c\le3\)

\(\frac{1}{4-\sqrt{ab}}+\frac{1}{4-\sqrt{bc}}+\frac{1}{4-\sqrt{ca}}\le\frac{1}{4-\frac{a+b}{2}}+\frac{1}{4-\frac{b+c}{2}}+\frac{1}{4-\frac{c+a}{2}}\)

\(=-\left(\frac{1}{\frac{a+b}{2}-4}+\frac{1}{\frac{b+c}{2}-4}+\frac{1}{\frac{c+a}{2}-4}\right)\le\frac{-\left(1+1+1\right)^2}{a+b+c-12}=\frac{-9}{3-12}=1\)

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

hơi phiền bn,bn có thẻ chỉ mik k ?

\(\frac{1}{a^2+b^2+2}+\frac{1}{b^2+c^2+2}+\frac{1}{c^2+a^2+2}\le\frac{3}{4}\)

\(\Leftrightarrow\frac{a^2+b^2}{a^2+b^2+2}+\frac{b^2+c^2}{b^2+c^2+2}+\frac{c^2+a^2}{c^2+a^2+2}\ge\frac{3}{2}\)

Áp dụng BĐT Cauchy - Schwarz ta có :

\(VT\ge\frac{\left(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}\right)^2}{2\left(a^2+b^2+c^2\right)+6}\)

\(\ge\frac{\sqrt{3\left(a^2b^2+b^2c^2+a^2c^2\right)}+2\left(a^2+b^2+c^2\right)}{a^2+^2+c^2}\)

\(\ge\frac{2\left(a^2+b^2+c^2\right)+ab+bc+ca}{a^2+b^2+c^2}\)

Ta cần chứng minh :

\(\frac{2\left(a^2+b^2+c^2\right)+ab+bc+ca}{a^2+b^2+c^2}\ge\frac{3}{2}\)

\(\Leftrightarrow\left(a+b+c\right)^2\ge0\) luôn đúng 

Chúc bạn học tốt !!!

hoang viet nhat copy nhớ ghi nguồn nha bạn:))Link 

Mà quan trọng là copy mà bạn có hiểu không là chuyện khác:) Bạn hãy giải thích tại sao:

\(\frac{\left(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}\right)^2}{2\left(a^2+b^2+c^2\right)+6}\ge\frac{\sqrt{3\left(a^2b^2+b^2c^2+c^2a^2\right)}+2\left(a^2+b^2+c^2\right)}{a^2+^2+c^2}\)

Mình có cách này,không chắc lắm:

\(VT=\frac{a}{a\left(a^2+bc+1\right)}+\frac{b}{b\left(b^2+ac+1\right)}+\frac{c}{c\left(c^2+ab+1\right)}\) (làm tắt,bạn tự hiểu nha)

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

\(\le\frac{1}{3}\left(\frac{1}{\sqrt[3]{a}}+\frac{1}{\sqrt[3]{b}}+\frac{1}{\sqrt[3]{c}}\right)\)

\(=\frac{1}{3}\left[\left(1+1+1\right)-\left(\frac{\sqrt[3]{a}-1}{\sqrt[3]{a}}+\frac{\sqrt[3]{b}-1}{\sqrt[3]{b}}+\frac{\sqrt[3]{c}-1}{\sqrt[3]{c}}\right)\right]\)

\(=1-\frac{1}{3}\left(\frac{\sqrt[3]{a}-1}{\sqrt[3]{a}}+\frac{\sqrt[3]{b}-1}{\sqrt[3]{b}}+\frac{\sqrt[3]{c}-1}{\sqrt[3]{c}}\right)\)

Áp dụng BĐT Cô si với biểu thức trong ngoặc:

\(=1-\frac{1}{3}\left(\frac{\sqrt[3]{a}-1}{\sqrt[3]{a}}+\frac{\sqrt[3]{b}-1}{\sqrt[3]{b}}+\frac{\sqrt[3]{c}-1}{\sqrt[3]{c}}\right)\)

\(\le1-\sqrt[3]{\left(\sqrt[3]{a}-1\right)\left(\sqrt[3]{b}-1\right)\left(\sqrt[3]{c-1}\right)}\le1^{\left(đpcm\right)}\)

Dấu "=" xảy ra khi a = b = c = 1

Ta c/m bđt sau: 

\(a^3+1\ge a^2+a\)

\(\Leftrightarrow a^3+1-a^2-a\ge0\Leftrightarrow a\left(a^2-1\right)-\left(a^2-1\right)\ge0\Leftrightarrow\left(a-1\right)^2\left(a+1\right)\ge0\)

\(\Rightarrow\frac{a}{a^3+a+1}\le\frac{a}{a^2+2a}=\frac{1}{a+2}\)

\(\Rightarrow\frac{a}{a^3+a+1}+\frac{b}{b^3+b+1}+\frac{c}{c^3+c+1}\le\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\)

Đặt \((a,b,c)\rightarrow(\frac{x}{y},\frac{y}{z},\frac{z}{x})\)

\(\Rightarrow\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}=\frac{y}{x+2y}+\frac{z}{y+2z}+\frac{x}{z+2x}=\frac{1}{2}\left(1-\frac{x}{x+2y}+1-\frac{y}{y+2z}+1-\frac{z}{z+2x}\right)=\frac{3}{2}-\frac{1}{2}\left(\frac{x^2}{x^2+2xy}+\frac{y^2}{y^2+2yz}+\frac{z^2}{z^2+2xy}\right)\)\(\le\frac{3}{2}-\frac{1}{2}\left(\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2+2xy+2yz+2zx}\right)=\frac{3}{2}-\frac{1}{2}.\frac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2}=1\)

Dấu bằng xảy ra khi a=b=c=1

BĐT <=> \(\frac{2}{a^2+2}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\le2\)

\(\Leftrightarrow1-\frac{a^2}{a^2+2}+1-\frac{b^2}{b^2+2}+1-\frac{c^2}{c^2+2}\le2\)

\(\Leftrightarrow\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}\ge1\)

Theo BĐT Svacxo:

\(VT\ge\frac{\left(a+b+c\right)^2}{a^2+b^2+c^2+6}=\frac{a^2+b^2+c^2+2\left(ab+bc+ca\right)}{a^2+b^2+c^2+6}=\frac{a^2+b^2+c^2+6}{a^2+b^2+c^2+6}=1\)

Vậy ta có đpcm.

P/s: Đúng ko ta?

Đặt: \(M=\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}=\Sigma_{cyc}\frac{a}{a^2+ab+bc+ca}\)

\(\Rightarrow M.\left(a+b+c\right)=3-\Sigma_{cyc}\frac{bc}{a^2+ab+bc+ca}\)

Đến đây t cần chứng minh:

 \(\frac{bc}{a^2+ab+bc+ca}+\frac{ca}{b^2+ab+bc+ca}+\frac{ab}{c^2+ab+bc+ca}\ge\frac{3}{4}\) (*)

Từ điều kiện ta có: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)

Đặt: \(\frac{1}{a}=x;\frac{1}{b}=y;\frac{1}{c}=z\left(x,y,z>0\right)\)

\(\Rightarrow x+y+z=1\)

(*) \(\Leftrightarrow\frac{x^2}{\left(x+y\right)\left(z+x\right)}+\frac{y^2}{\left(x+y\right)\left(y+z\right)}+\frac{z^2}{\left(y+z\right)\left(z+x\right)}\ge\frac{3}{4}\)

Theo Cô-si: \(\frac{x^2}{\left(x+y\right)\left(z+x\right)}+\frac{9}{16}\left(x+y\right)\left(z+x\right)\ge\frac{3}{2}x\)

Nhứng phần kia tương tự

\(\Rightarrow\Sigma_{cyc}\frac{x^2}{\left(x+y\right)\left(z+x\right)}\ge\frac{3}{2}\left(x+y+z\right)-\frac{9}{16}\left[\left(x+y+z\right)^2+\left(xy+yz+zx\right)\right]\ge\frac{3}{4}\)

Lần trước làm không đúng hy vọng bây giờ gỡ lại được

nub

Bạn suy ra dòng 8 mk chưa hiểu, giải kĩ cho mk đc ko