\(A=\frac{x}{1+y^2}+\frac{y}{1+z^2}+\frac{z}{1+x^2}=x\left(1-\frac{y^2}{1+y^2}\right)+y\left(1-\frac{z^2}{1+z^2}\right)+z\left(1-\frac{x^2}{1+x^2}\right)\)

\(\Rightarrow A\ge x\left(1-\frac{y}{2}\right)+y\left(1-\frac{z}{2}\right)+z\left(1-\frac{x}{2}\right)=\left(x+y+z\right)-\frac{xy+yz+zx}{2}\ge3-\frac{\frac{9}{3}}{2}=\frac{3}{2}\)

Dau '=' xay ra khi \(x=y=z=1\)

Vay \(A_{min}=\frac{3}{2}\)khi \(x=y=z=1\)

Cauchy ngược dấu:v

\(A\ge x\left(1-\frac{y}{2}\right)+y\left(1-\frac{z}{2}\right)+z\left(1-\frac{x}{2}\right)\)

\(=x+y+z-\frac{xy+yz+zx}{2}\ge3-\frac{\left(x+y+z\right)^2}{6}=\frac{3}{2}\)

Đẳng thức xảy ra khi x = y = z = 1.

P/s: Ko chắc~

\(A\ge\frac{1}{3}\left(x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}\right)^2\ge\frac{1}{3}\left(x+y+z+\frac{9}{x+y+z}\right)^2=\frac{100}{3}\)

Dấu "=" xảy ra khi \(x=y=z=\frac{1}{3}\)

\(\frac{17}{3}\) đúng k?

Áp dụng Bunhia.

\(\left(x+y+z\right)^2\le\left(1^2+1^2+1^2\right)\left(x^2+y^2+z^2\right)=3.3=9\)

=> \(0< x+y+z\le3\)

Có: \(P=\frac{x^2+1}{x}+\frac{y^2+1}{y}+\frac{z^2+1}{z}-\frac{1}{x+y+z}\)

\(=\frac{x^2-2x+1}{x}+\frac{y^2-2y+1}{y}+\frac{z^2-2z+1}{z}-\frac{1}{x+y+z}+6\)

\(=\frac{\left(x-1\right)^2}{x}+\frac{\left(y-1\right)^2}{y}+\frac{\left(z-1\right)^2}{z}-\frac{1}{x+y+z}+6\)

\(\ge\frac{\left(x+y+z-3\right)^2}{x+y+z}-\frac{1}{x+y+z}+6=\frac{\left(x+y+z-3\right)^2-1}{x+y+z}+6\)

\(\ge\frac{0-1}{3}+6=\frac{17}{3}\)

"=" xảy ra <=> \(x+y+z=3;x=y=z\Leftrightarrow x=y=z=1\)

Vậy min P = 17/3 <=> x = y = z =1.

\(P=\frac{x^2+1}{x}+\frac{y^2+1}{y}+\frac{z^2+1}{z}-\frac{1}{x+y+z}\)

\(=x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\frac{1}{x+y+z}\)

\(\ge x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-\frac{1}{9}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)

\(=x+y+z+\frac{8x}{9}+\frac{8y}{9}+\frac{8z}{9}\)

Có BĐT phụ \(a+\frac{8}{9a}\ge\frac{a^2+33}{18}\)

\(\Leftrightarrow\frac{9a^2+8}{9a}\ge\frac{a^2+33}{18}\)

\(\Leftrightarrow162a^2+144-9a^3-297a\ge0\)

\(\Leftrightarrow-a^3+18a^2-33a+16\ge0\)

\(\Leftrightarrow\left(a-1\right)^2\left(16-a\right)\ge0\left(OK\right)\)

\(\Rightarrow P\ge\frac{x^2+y^2+z^2+99}{18}=\frac{17}{3}\)

Dấu "=" xảy ra tại x=y=z=1

\(\frac{\left(x+y+z-3\right)^2-1}{x+y+z}+6\ge\frac{0-1}{x+y+z}+6\ge\frac{0-1}{3}+6\)

Chú ý: \(\left(x+y+z-3\right)^2\ge0\) với mọi x, y, z.

a/ \(\frac{2x+1}{\sqrt{x^2+2}}+\left(x+1\right)\left(\sqrt{1+\frac{2x+1}{x^2+2}}-1\right)+2x+1=0\)

\(\Leftrightarrow\frac{2x+1}{\sqrt{x^2+2}}+\frac{\left(x+1\right)\left(2x+1\right)}{\sqrt{1+\frac{2x+1}{x^2+2}}+1}+2x+1=0\)

\(\Leftrightarrow\left(2x+1\right)\left(\frac{1}{\sqrt{x^2+2}}+\frac{x+1}{\sqrt{1+\frac{2x+1}{x^2+2}}+1}+1\right)=0\)

\(\Rightarrow x=-\frac{1}{2}\)

b/ \(Q\ge\frac{\left(x+y+z\right)^2}{xyz\left(x+y+z\right)}+\frac{\left(x^3+y^3+z^3\right)^2}{xy+yz+zx}\ge\frac{x+y+z}{xyz}+\frac{\left(x^2+y^2+z^2\right)^3}{\left(x+y+z\right)^2}\)

\(Q\ge\frac{27\left(x+y+z\right)}{\left(x+y+z\right)^3}+\frac{\left(x+y+z\right)^6}{27\left(x+y+z\right)^2}=\frac{27}{\left(x+y+z\right)^2}+\frac{\left(x+y+z\right)^4}{27}\)

\(Q\ge\frac{27}{64\left(x+y+z\right)^2}+\frac{27}{64\left(x+y+z\right)^2}+\frac{\left(x+y+z\right)^4}{27}+\frac{837}{32\left(x+y+z\right)^2}\)

\(Q\ge3\sqrt[3]{\frac{27^2\left(x+y+z\right)^4}{64^2.27\left(x+y+z\right)^4}}+\frac{837}{32.\left(\frac{3}{2}\right)^2}=\frac{195}{16}\)

"=" \(\Leftrightarrow x=y=z=\frac{1}{2}\)

Nguyễn Trúc Giang, Duy Khang, Vũ Minh Tuấn, Võ Hồng Phúc, tth, No choice teen, Phạm Lan Hương,

Nguyễn Lê Phước Thịnh, @Nguyễn Việt Lâm, @Akai Haruma

giúp em vs ạ! Cần trước 5h chiều nay ạ

Thanks nhiều

3, \(P=a+b+\frac{1}{2a}+\frac{2}{b}\)

=\(\left(\frac{1}{2a}+\frac{a}{2}\right)+\left(\frac{b}{2}+\frac{2}{b}\right)+\frac{a+b}{2}\)

AD bđt cosi vs hai số dương có:

\(\frac{1}{2a}+\frac{a}{2}\ge2\sqrt{\frac{1}{2a}.\frac{a}{2}}=2\sqrt{\frac{1}{4}}=1\)

\(\frac{b}{2}+\frac{2}{b}\ge2\sqrt{\frac{b}{2}.\frac{2}{b}}=2\)

Có \(\frac{a+b}{2}\ge\frac{3}{2}\) (vì a+b \(\ge3\))

=> \(P=\left(\frac{1}{2a}+\frac{a}{2}\right)+\left(\frac{b}{2}+\frac{2}{b}\right)+\frac{a+b}{2}\ge1+2+\frac{3}{2}\)

<=> P \(\ge4.5\)

Dấu "=" xảy ra <=>\(\left\{{}\begin{matrix}\frac{1}{2a}=\frac{a}{2}\\\frac{b}{2}=\frac{2}{b}\\a+b=3\end{matrix}\right.\) <=>\(\left\{{}\begin{matrix}a^2=1\\b^2=4\\a+b=3\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}a=1\\b=2\\a+b=3\end{matrix}\right.\)

=> a=2,b=3

Vậy minP=4.5 <=>a=1,b=2