Dăm 3 vài phương pháp sos:)

\(P=\Sigma_{cyc}\frac{\left(x-y\right)^2\left(x+2y\right)}{y^2}+\left(x+y+z\right)\ge x+y+z=3\)

Đẳng thức xảy ra khi x = y = z = 1

ĐKXĐ : ....

PT \(\Leftrightarrow\sqrt{-x^2+4x-3}-1+\sqrt{-2x^2+8x+1}-3=x\left(x^2-4x+4\right)\)

\(\Leftrightarrow\frac{-x^2+4x-4}{\sqrt{-x^2+4x-3}+1}+\frac{-2x^2+8x-8}{\sqrt{-2x^2+8x+1}+3}=x\left(x-2\right)^2\)

\(\Leftrightarrow\frac{\left(x-2\right)^2}{\sqrt{-x^2+4x-3}+1}+\frac{2\left(x-2\right)^2}{\sqrt{-2x^2+8x+1}+3}+x\left(x-2\right)^2=0\)

\(\Leftrightarrow\left(x-2\right)^2\left(\frac{1}{\sqrt{-x^2+4x-3}+1}+\frac{2}{\sqrt{-2x^2+8x+1}+3}+x\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}\left(x-2\right)^2=0\\\frac{1}{\sqrt{-x^2+4x-3}+1}+\frac{2}{\sqrt{-2x^2+8x+1}+3}+x=0\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=0\\\frac{1}{\sqrt{-x^2+4x-3}+1}+\frac{2}{\sqrt{-2x^2+8x+1}+3}+x>0\left(loai\right)\end{cases}}\)

Bạn Thanh Tùng DZ ơi sao trường hợp 2 lại loại vậy

Chưa có điều kiện của x mà

\(a^2+b^2+1\ge ab+a+b\)

\(\Leftrightarrow2a^2+2b^2+2-2ab-2a-2b\ge0\)

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

\(\Leftrightarrow\left(a-b\right)^2+\left(a-1\right)^2+\left(b-1\right)^2\ge0\left(true!!\right)\)

Dấu "=" xảy ra tại a=b=1

Xét hiệu \(A=\left(a^2+b^2+1\right)-\left(ab+a+b\right)\)

\(=a^2+b^2+1-ab-a-b\)

\(\Rightarrow2A=2a^2+2b^2+2-2ab-2a-2b\)

\(=\left(a^2-2ab+b^2\right)+\left(a^2-2a+1\right)+\left(b^2-2b+1\right)\)

\(=\left(a-b\right)^2+\left(a-1\right)^2+\left(b-1\right)^2\ge0\)

\(\Rightarrow2A\ge0\Leftrightarrow A\ge0\)

Vậy \(a^2+b^2+1\ge ab+a+b\left(đpcm\right)\)

Dấu " = " xảy ra \(\Leftrightarrow\hept{\begin{cases}a-b=0\\a-1=0\\b-1=0\end{cases}}\Leftrightarrow a=b=1\)

Ta có : \(ab+bc+ca=2abc\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2\)

Đặt \(\hept{\begin{cases}x=\frac{1}{a}\\y=\frac{1}{b}\\z=\frac{1}{c}\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}x+y+z=2\\P=\frac{x^3}{\left(2-x\right)^2}+\frac{y^3}{\left(2-y\right)^3}+\frac{z^3}{\left(2-z\right)^2}\end{cases}}\)

Áp dụng bất đẳng thức Cauchy - Schwarz 

\(\Rightarrow\frac{x^3}{\left(2-x\right)^2}+\frac{2-x}{8}+\frac{2-x}{8}\ge3\sqrt[3]{\frac{x^3}{64}}=\frac{3x}{4}\)

Tương tự ta có :

\(\hept{\begin{cases}\frac{y^3}{\left(2-y\right)^2}+\frac{2-y}{8}+\frac{2-y}{8}\ge\frac{3y}{4}\\\frac{z^3}{\left(2-z\right)^2}+\frac{2-z}{8}+\frac{2-z}{8}\ge\frac{3z}{8}\end{cases}}\)

\(\Rightarrow P+\frac{12-2\left(x+y+z\right)}{8}\ge\frac{3}{4}\left(x+y+z\right)\)

\(\Rightarrow P\ge\frac{1}{12}\)

Dấu " = " xảy ra khi \(x=y=z=\frac{2}{3}\)

Hệ \(\Leftrightarrow\hept{\begin{cases}\left(x+y\right)\left(x-2y\right)^2=\left(x-2y\right)^2\\\sqrt{x-2y}+\sqrt{3x+2y}=4x-4\end{cases}.}\)

\(\Leftrightarrow\hept{\begin{cases}\left(x-2y\right)^2\left(x+y-1\right)=0\\\sqrt{x-2y}+\sqrt{3x+2y}=4x-4\end{cases}}\)

Đến đây thì đơn giản rồi, tự làm nhé

\(P=\frac{\left(a+b\right)^2+ab}{\sqrt{ab}\left(a+b\right)}=\frac{a+b}{\sqrt{ab}}+\frac{\sqrt{ab}}{a+b}=\frac{3\left(a+b\right)}{4\sqrt{ab}}+\frac{a+b}{4\sqrt{ab}}+\frac{\sqrt{ab}}{a+b}\)

\(\Rightarrow P\ge\frac{3.2\sqrt{ab}}{4\sqrt{ab}}+2\sqrt{\frac{a+b}{4\sqrt{ab}}.\frac{\sqrt{ab}}{a+b}}=\frac{3}{2}+1=\frac{5}{2}\)

\(\Rightarrow P_{min}=\frac{5}{2}\) khi a=b