Ta có : 

A = \(\sqrt{x}+x\)

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

Ta có : \(\sqrt{x}\ge0\)\(\Rightarrow\)\(\left(\sqrt{x}+\frac{1}{2}\right)^2\ge\frac{1}{4}\)

Dấu " = " xảy ra \(\Leftrightarrow\)x = 0 

A = \(\left(\sqrt{x}+\frac{1}{2}\right)^2-\frac{1}{4}\ge0\)

Dấu " = " xảy ra \(\Leftrightarrow\)x = 0 

Vậy giá trị nhỏ nhất của A = 0 \(\Leftrightarrow\)x = 0

                     Giải

Ta có :\(A=\sqrt{x}+x\)

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

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

Ta có : \(\sqrt{x}\ge0\Rightarrow\left(\sqrt{x}+\frac{1}{2}\right)^2\ge4\)

\(\Rightarrow A=\left(\sqrt{x}+\frac{1}{2}\right)^2\ge0\)

Dấu " = " xảy ra \(\Leftrightarrow\)x = 0

Vậy giá trị nhỏ nhất của A là 0

Với mọi a,b ta có : ( a - b )² \(\ge\)0 

=> a² + b² \(\ge\)2ab => 2 . ( a² + b² ) \(\ge\)( a + b )² = 1

=> a² + b² \(\ge\)\(\frac{1}{2}\)

Dấu " = " xảy ra <=> a = b = \(\frac{1}{2}\)

\(a,\Leftrightarrow x^2+y^2+z^2-2xy+2xz-2yz\ge0\)

\(\Leftrightarrow\left(x-y+z\right)^2\ge0\left(đúng\right)\)

\(b,\Leftrightarrow4a^2+b^2-4ab\ge0\)

\(\Leftrightarrow\left(2a\right)^2+b^2-4ab\ge0\)

\(\Leftrightarrow\left(2a-b\right)^2\ge0\left(đúng\right)\)

P/s: chưa chắc lắm :(

PTĐTTNT?

1.Đặt \(a^2+a=t\)

\(\Rightarrow\left(a^2+a\right)\left(a^2+a+1\right)-2\)

\(=t\left(t+1\right)-2\)

\(=t^2+t-2\)

\(=t^2+2t-\left(t+2\right)\)

\(=t\left(t+2\right)-\left(t+2\right)\)

\(=\left(t+2\right)\left(t-1\right)\)

Sửa đề: 

\(x^4+2011x^2+2010x+2011\)

\(=\left(x^4-x\right)+2011x^2+2011x+2011\)

\(=x\left(x^3-1\right)+2011\left(x^2+x+1\right)\)

\(=x\left(x-1\right)\left(x^2+x+1\right)+2011\left(x^2+x+1\right)\)

\(=\left(x^2+x+1\right)\left(x^2-x+2011\right)\)

3. \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-120\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-120\)

Đặt \(x^2+5x+4=t\)

\(\Rightarrow\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-120\)

\(=t\left(t+2\right)-120\)

\(=t^2+2t+1-121\)

\(=\left(t+1\right)^2-11^2\)

\(=\left(t+1-11\right)\left(t+1+11\right)\)

\(=\left(t-10\right)\left(t+12\right)\)

\(=\left(x^2+5x-6\right)\left(x^2+5x+16\right)\)

\(=\left[\left(x^2-x\right)+\left(6x-6\right)\right]\left(x^2+5x+16\right)\)

\(=\left[x.\left(x-1\right)+6\left(x-1\right)\right]\left(x^2+5x+16\right)\)

\(=\left(x-1\right)\left(x+6\right)\left(x^2+5x+16\right)\)

4. \(\left(x^2+x+4\right)^2+8x\left(x^2+x+1\right)+15x^2\)

\(=\left(x^2+x+4\right)^2+2.\left(x^2+x+1\right).4x+\left(4x\right)^2-x^2\)

\(=\left(x^2+x+4+4x\right)^2-x^2\)

\(=\left(x^2+4+5x-x\right)\left(x^2+5x+x+4\right)\)

\(=\left(x^2+4x+4\right)\left(x^2+6x+4\right)\)

\(=\left(x+2\right)^2\left[\left(x^2+2.x.3+3^2\right)-\left(\sqrt{5}\right)^2\right]\)

\(=\left(x+2\right)^2\left[\left(x+3\right)^2-\left(\sqrt{5}\right)^2\right]\)

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

Ta có: \(\left(x-\frac{1}{x}\right):\left(x+\frac{1}{x}\right)=n\Rightarrow\frac{x^2-1}{x}:\frac{x^2+1}{x}=n\Rightarrow\frac{x^2-1}{x^2+1}=n\)

\(\Rightarrow x^2-1=n\left(x^2+1\right)=nx^2+n\Rightarrow x^2-nx^2=n+1\Rightarrow x^2\left(1-n\right)=n+1\Rightarrow x^2=\frac{n+1}{1-n}\left(n\ne1\right)\)

THay vào V ta được: \(V=\left(\frac{n+1}{1-n}-\frac{1}{\frac{n+1}{1-n}}\right):\left(\frac{n+1}{1-n}+\frac{1}{\frac{n+1}{1-n}}\right)=\left(\frac{n+1}{1-n}-\frac{1-n}{n+1}\right):\left(\frac{n+1}{1-n}+\frac{1-n}{n+1}\right)\)

\(=\frac{\left(n+1\right)^2-\left(1-n\right)^2}{\left(n+1\right)\left(1-n\right)}:\frac{\left(n+1\right)^2+\left(1-n\right)^2}{\left(n+1\right)\left(1-n\right)}=\frac{n^2+2n+1-1+2n-n^2}{n^2+2n+1+1-2n+n^2}\)

\(=\frac{4n}{2n^2+2}=\frac{4n}{2\left(n^2+1\right)}=\frac{2n}{n^2+1}\)