\(P\ge\dfrac{1}{2}\left(\dfrac{1}{x-2}+\dfrac{1}{3-x}\right)^2+\dfrac{4}{\left(x-2+3-x\right)^2}=\dfrac{1}{2}\left(\dfrac{1}{x-2}+\dfrac{1}{3-x}\right)^2+4\)

\(P\ge\dfrac{1}{2}\left(\dfrac{4}{x-4+3-x}\right)^2+4=12\)

Dấu "=" xảy ra khi \(x-2=3-x\Rightarrow x=\dfrac{5}{2}\)

Đặt \(x+3=t\ne0\Rightarrow x=t-3\)

\(A=\dfrac{\left(t+2\right)\left(t-4\right)}{t^2}=\dfrac{t^2-2t-8}{t^2}=-\dfrac{8}{t^2}-\dfrac{2}{t}+1=-8\left(\dfrac{1}{t}+\dfrac{1}{8}\right)^2+\dfrac{9}{8}\le\dfrac{9}{8}\)

\(A_{max}=\dfrac{9}{8}\) khi \(t=-8\) hay \(x=-11\)

\(y=x+\dfrac{1}{x}-5\ge2\sqrt{\dfrac{x}{x}}-5=-3\)

\(y_{min}=-3\) khi \(x=1\)

\(y=4x^2+\dfrac{1}{2x}+\dfrac{1}{2x}-4\ge3\sqrt[3]{\dfrac{4x^2}{2x.2x}}-4=-1\)

\(y_{min}=-1\) khi \(x=\dfrac{1}{2}\)

\(y=x+\dfrac{4}{x}\Rightarrow y'=1-\dfrac{4}{x^2}=0\Rightarrow x=-2\)

\(y\left(-2\right)=-4\Rightarrow\max\limits_{x>0}y=-4\) khi \(x=-2\)

\(A=\left|x-1\right|+\left|x-2\right|+\left|x-3\right|+\left|x-4\right|\)

\(A=\left|1-x\right|+\left|x-4\right|+\left|2-x\right|+\left|x-3\right|\)

Ta có: \(\left|1-x\right|+\left|x-4\right|\ge\left|1-x+x-4\right|=3\)

           \(\left|2-x\right|+\left|x-3\right|\ge\left|2-x+x-3\right|=1\) 

=> \(\left|1-x\right|+\left|x-4\right|+\left|2-x\right|+\left|x-3\right|\ge3+1=4\)

=> \(A\ge4\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(1-x\right)\left(x-4\right)\ge0\\\left(2-x\right)\left(x-3\right)\ge0\end{cases}}\)

                        \(\Leftrightarrow\hept{\begin{cases}1\le x\le3\\2\le x\le4\end{cases}}\)

                        \(\Leftrightarrow2\le x\le3\)

Vậy \(A_{min}=4\Leftrightarrow2\le x\le3\)

A=(x^2+5x-6)(x^2+5x+6)

=(x^2+5x)^2-36>=-36

Dấu = xảy ra khi x=0 hoặc x=-5

Giấ trị nhỏ nhất là 8

GTNN = 8 đạt khi   $t=0\iff x=2$

Áp dụng BĐT \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\):

\(A=\left|x-3\right|+\left|x-1\right|+\left|x+1\right|+\left|x+3\right|\)

\(=\left|3-x\right|+\left|x+3\right|+\left|1-x\right|+\left|x+1\right|\)

\(\ge\left|3-x+x+3\right|+\left|1-x+x+1\right|=8\)

\(minA=8\Leftrightarrow\left\{{}\begin{matrix}\left(3-x\right)\left(x+3\right)\ge0\\\left(1-x\right)\left(x+1\right)\ge0\end{matrix}\right.\Leftrightarrow-1\le x\le1\)