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Ta có BĐT phụ : \(a^3+b^3\ge ab\left(a+b\right)\)
\(\Rightarrow a^3+b^3+1\ge ab\left(a+b\right)+abc=ab\left(a+b+c\right)\)
\(\Rightarrow\frac{1}{a^3+b^3+1}\le\frac{1}{ab\left(a+b+c\right)}\)
Tương tự : ...
\(\Rightarrow\frac{1}{a^3+b^3+1}+\frac{1}{b^3+c^3+1}+\frac{1}{a^3+c^3+1}\le\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\frac{1}{a+b+c}\)
\(=\frac{a+b+c}{abc}.\frac{1}{a+b+c}=\frac{1}{abc}=1\)
BĐT đã được c/m.
Dấu "=" xảy ra khi a = b = c = 1
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
Đặt \(a=\frac{x}{y},b=\frac{y}{z},c=\frac{z}{x}\) là ra bạn KK
ta có : \(\frac{1}{a^3\left(b+c\right)}+\frac{1}{b^3\left(a+c\right)}+\frac{1}{c^3\left(a+b\right)}=\frac{\frac{1}{a^2}}{a\left(b+c\right)}+\frac{\frac{1}{b^2}}{b\cdot\left(a+c\right)}+\frac{\frac{1}{c^2}}{c\left(a+b\right)}\)(1)
dùng Svaxo ta có (1) <=>\(\frac{\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)^2}{a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)}=\frac{ab+bc+ca}{2}>=\frac{3a^2b^2c^2}{2}=\frac{3}{2}\)(côsi )
với x,y >0 ta có : \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\Rightarrow\frac{1}{x+y}\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}\right)..\)
Áp dụng bất đẳng thức trên được:
\(\frac{1}{ab+a+2}=\frac{1}{\left(ab+1\right)+\left(a+1\right)}\le\frac{1}{4}\left(\frac{1}{ab+1}+\frac{1}{a+1}\right)=\frac{1}{4}\left(\frac{abc}{ab+abc}+\frac{1}{a+1}\right)=\frac{1}{4}\left(\frac{c}{c+1}+\frac{1}{a+1}\right)\left(1\right).\)( vì abc = 1 )
Chứng minh tương tự ta được : \(\frac{1}{bc+b+2}\le\frac{1}{4}\left(\frac{a}{a+1}+\frac{1}{b+1}\right)\left(2\right).\)
\(\frac{1}{ac+c+2}\le\frac{1}{4}\left(\frac{b}{b+1}+\frac{1}{c+1}\right)\left(3\right).\)
Cộng vế với vế các BĐT (1), (2) và (3) ta được :
\(P\le\frac{1}{4}\left(\frac{a}{a+1}+\frac{1}{a+1}+\frac{b}{b+1}+\frac{1}{b+1}+\frac{c}{c+1}+\frac{1}{c+1}\right)=\frac{3}{4}.\)( đpcm )
dấu " = " xẩy ra khi a = b = c = 1
Đặt \(\left(a;b;c\right)\rightarrow\left(x^3;y^3;z^3\right)\Rightarrow xyz=1\)
Ta có:
\(\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\)
\(=\frac{1}{x^3+y^3+1}+\frac{1}{y^3+z^3+1}+\frac{1}{z^3+x^3+1}\left(1\right)\)
Áp dụng BĐT phụ \(x^3+y^3\ge xy\left(x+y\right)\)
\(\Rightarrow\left(1\right)\le\frac{1}{xy\left(x+y\right)+xyz}+\frac{1}{yz\left(y+z\right)+xyz}+\frac{1}{zx\left(z+x\right)+xyz}\)
\(=\frac{1}{xy\left(x+y+z\right)}+\frac{1}{yz\left(x+y+z\right)}+\frac{1}{zx\left(x+y+z\right)}\)
\(=\frac{z}{xyz\left(x+y+z\right)}+\frac{x}{xyz\left(x+y+z\right)}+\frac{z}{xyz\left(x+y+z\right)}\)
\(=\frac{x+y+z}{xyz\left(x+y+z\right)}=1\)
Dấu "=" xảy ra tại \(x=y=z=1\) hay \(a=b=c=1\)
Nhầm dòng thứ 3 dưới lên ạ:(
\(\frac{z}{xyz\left(x+y+z\right)}+\frac{x}{xyz\left(x+y+z\right)}+\frac{y}{xyz\left(x+y+z\right)}\) mới đúng nha !