Áp dụng cosi ta có \(a.a.a.b.b\le\frac{3a^5+2b^5}{5};b.b.b.a.a\le\frac{3b^5+2a^5}{5}\)

=> \(a^5+b^5\ge a^2b^2\left(a+b\right)\)

Khi đó

\(VT\le\frac{1}{ab\sqrt{a+b}}+\frac{1}{bc\sqrt{b+c}}+\frac{1}{ac\sqrt{a+c}}\)

Áp dụng BĐT buniacoxki  ta có :

\((\frac{1}{ab\sqrt{a+b}}+\frac{1}{bc\sqrt{b+c}}+\frac{1}{ac\sqrt{a+c}})^2\le\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\left(\frac{1}{b^2\left(a+b\right)}+\frac{1}{c^2\left(b+c\right)}+...\right)\)

Mà 1/a^2+1/b^2+1/c^2=1(giả thiết)

=> \(VT\le VP\)(ĐPCM)

Dấu bằng xảy ra khi a=b=c=can(3)

hay quá 

đang linh tinh ngáo ak 

câu hỏi mà cũng đăng nội quy 

trẻ trâu lắm thiệt

Thi đấu gì vậy bn ? 

\(a,|x+1|+|x-1|=4\)\((*)\)

• \(TH_1:\hept{\begin{cases}x+1\ge0\\x-1\ge0\end{cases}}\Leftrightarrow x\ge1\)

Khi đó pt \((*)\) \(\Leftrightarrow x+1+x-1=4\Leftrightarrow x=2\left(tm\right)\)

• \(TH_2:\hept{\begin{cases}x+1\ge0\\x-1< 0\end{cases}}\Leftrightarrow-1\le x< 1\)

Khi đó pt \((*)\) \(\Leftrightarrow x+1+1-x=4\Leftrightarrow0=2\left(vl\right)\)

• \(TH_3:\hept{\begin{cases}x+1< 0\\x-1\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x< -1\\x\ge1\end{cases}}\left(l\right)\)
• \(TH_4:\hept{\begin{cases}x+1< 0\\x-1>0\end{cases}}\Leftrightarrow x< -1\)

Khi đó pt \((*)\) \(\Leftrightarrow-x-1+1-x=4\)

\(\Leftrightarrow x=-2\left(tm\right)\)

Vậy pt có 2 \(n_0\) \(x=\pm2\)

\(b,\frac{|2x-1|}{x+1}=\frac{1}{2}\left(Đkxđ:x\ne-1\right)\)

\(\Rightarrow2|2x-1|=x+1\)

• \(TH_1:x\ge\frac{1}{2}\Leftrightarrow2\left(2x-1\right)=x+1\)

\(\Leftrightarrow4x-2=x+1\)

\(\Leftrightarrow x=1\left(tm\right)\)

• \(TH_2:x< \frac{1}{2}\Leftrightarrow2\left(1-2x\right)=x+1\)

\(\Leftrightarrow2-4x=x+1\)

\(\Leftrightarrow x=\frac{1}{5}\left(l\right)\)

Vậy pt có 1 \(n_0\) \(x=1\)

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

    \(\ge a-\frac{ab^2}{2b}+b-\frac{bc^2}{2c}+c-\frac{ca^2}{2c}\) (AM-GM)

      \(\ge a-\frac{ab}{2}+b-\frac{bc}{2}+c-\frac{ac}{2}\ge\left(a+b+c\right)-\frac{\left(a+b+c\right)^2}{6}\ge3-\frac{3}{2}=\frac{3}{2}\)

Vay MinP=3/2 dau = xay ra khi a=b=c=1