Đặt \(\left(\sqrt{a};\sqrt{b};\sqrt{c}\right)\rightarrow\left(x;y;z\right)\)\(\Rightarrow\)\(x^2+y^2+z^2=4\)

\(P=\frac{x^3}{x+3y}+\frac{y^3}{y+3z}+\frac{z^3}{z+3x}=\frac{x^4}{x^2+3xy}+\frac{y^4}{y^2+3yz}+\frac{z^4}{z^2+3zx}\)

\(\ge\frac{\left(x^2+y^2+z^2\right)^2}{x^2+y^2+z^2+3\left(xy+yz+zx\right)}\ge\frac{\left(x^2+y^2+z^2\right)^2}{x^2+y^2+z^2+3\left(x^2+y^2+z^2\right)}=\frac{4^2}{4+3.4}=1\)

Dấu "=" xảy ra \(\Leftrightarrow\)\(a=b=c=\frac{2}{\sqrt{3}}\)

à nhầm, \(a=b=c=\frac{4}{3}\) nhé 

a: Để A là số nguyên thì

x^3-2x^2+4 chia hết cho x-2

=>\(x-2\in\left\{1;-1;2;-2;4;-4\right\}\)

=>\(x\in\left\{3;1;4;0;6;-2\right\}\)

b: Để B là số nguyên thì

\(3x^3-x^2-6x^2+2x+9x-3+2⋮3x-1\)

=>\(3x-1\in\left\{1;-1;2;-2\right\}\)

=>\(x\in\left\{\dfrac{2}{3};0;1;-\dfrac{1}{3}\right\}\)

\(a,P=\frac{x+2}{x-2}+\frac{x}{x+2}-\frac{4}{x^2-4}\)

\(P=\frac{\left(x+2\right)^2}{\left(x-2\right)\left(x+2\right)}+\frac{x\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}-\frac{4}{\left(x-2\right)\left(x+2\right)}\)

\(P=\frac{x^2+4x+4+x^2-2x-4}{x^2-4}\)

\(P=\frac{2x^2+2x}{x^2-4}\)

\(P=\frac{2x^2+2x}{x^2-4}\)               (1)

\(b,x^2-3x=0\)

\(\Leftrightarrow x\left(x-3\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=0\left(ktm\right)\\x=3\left(tm\right)\end{cases}}\)

thay vào (1) ta có : 

\(P=\frac{2\cdot3^2+2\cdot3}{3^2-4}=\frac{24}{5}\)

\(A=\left(\frac{2X-1}{x^2-4}+\frac{x+2}{x^2-x-2}\right):\frac{x-2}{x^2+3x+2}ĐK:x\ne\left\{2,-2,-1\right\}\)

a)  \(A=\left[\frac{\left(2x-1\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x+2}{\left(x+1\right)\left(x-2\right)}\right]:\frac{x-2}{\left(x+2\right)\left(x+1\right)}\)

\(A=\left[\frac{\left(2x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x+2\right)\left(x+1\right)}\frac{\left(x+2\right)\left(x+2\right)}{\left(x+1\right)\left(x-2\right)\left(x+2\right)}\right].\frac{\left(x+2\right)\left(x+1\right)}{x-2}\)

\(A=\frac{2x^2+x-1+x^2+4x.4}{\left(x-2\right)\left(x+2\right)\left(x+1\right)}.\frac{\left(x+2\right)\left(x+1\right)}{\left(x-2\right)}\)

\(A=\frac{3x^2+5x+3}{\left(x-2\right)\left(x+2\right)\left(x+1\right)}.\frac{\left(x+2\right)\left(x+1\right)}{\left(x-2\right)}\)

\(A=\frac{3x^2+5x+3}{\left(x-2\right)^2}\)

Ta có :\(3x^2+5x+3\)

\(=3\left(x^2+\frac{5}{3}x+1\right)\)

\(=3\left[x^2+2.\frac{5}{6}x+\frac{25}{36}+\frac{9}{36}\right]\)

\(=3\left[\left(x+\frac{5}{6}\right)^2+\frac{9}{36}\right]>0\)

Mà \(\left(x-2\right)^2>0\)

\(\Rightarrow A>0\left(dpcm\right)\)

\(b,A=11\Leftrightarrow\frac{3x^2+5x+3}{\left(x-2\right)^2}=11\)

\(\Rightarrow3x^2+5x+3=11.\left(x-2\right)^2\)

\(\Rightarrow3x^2+5x+3=11.\left(x^2-4x+4\right)\)

\(\Rightarrow8x^2-49x+41=0\)

\(\Rightarrow8x^2-8x-41x+41=0\)

\(\Rightarrow8x\left(x-1\right)-41\left(x-1\right)=0\)

\(\Rightarrow\left(8x-41\right)\left(x-1\right)=0\)

\(\Leftrightarrow\hept{\begin{cases}8x-41=0\\x-1=0\end{cases}\Rightarrow\hept{\begin{cases}x=\frac{41}{8}\\x=1\end{cases}}}\)(Thỏa mãn)