a) \(M=\frac{x^2+\sqrt{x}}{x-\sqrt{x}+1}+1\)\(-\frac{2x+\sqrt{x}}{\sqrt{x}}\)

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

\(=\sqrt{x}\left(\sqrt{x}+1\right)-2\sqrt{x}\)

\(=x+\sqrt{x}-2\sqrt{x}=x-\sqrt{x}\)

HD:Có P=2x+1/x^2=x+x+1/x ^2>=3 căn bậc 3 (x.x.1/x^2)=3.(x>0)

MinP=3<=>x=1/x^2<=>x=1.

1.(√x -2)^2 ≥ 0 --> x -4√x +4 ≥ 0 --> x+16 ≥ 12 +4√x --> (x+16)/(3+√x) ≥4 
--> Pmin=4 khi x=4

2. Đặt \(\sqrt{x^2-4x+5}=t\ge1\)1

=> M=2x²-8x+\(\sqrt{x^2-4x+5}\)+6=2(t²-5)+t+6

<=> M=2t²+t-4\(\ge\)2.1²+1-4=-1

M_min=-1 khi t=1 hay x=2

1.  x≥1 <=> \(\frac{1}{x}\le1\Leftrightarrow\frac{1}{x}+1\le2\Leftrightarrow A\le2\Rightarrow MaxA=2\Leftrightarrow x=1\)

2. Áp dụng bđt cosi cho x>0. ta có: \(x+\frac{1}{x}\ge2\sqrt{x.\frac{1}{x}}=2\Leftrightarrow P\ge2\Rightarrow MinP=2\Leftrightarrow x=\frac{1}{x}\Leftrightarrow x=1\)

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

áp dụng cosi cho 2 số dương ta có: \(x+1+\frac{4}{x+1}\ge2\sqrt{x+1.\frac{4}{x+1}}=2\Leftrightarrow A+1\ge2\Rightarrow A\ge3\Rightarrow MinA=3\Leftrightarrow x+1=\frac{4}{x+1}\Leftrightarrow x=1\)

Vì \(A=\frac{x^2-2x+2014}{\left(x+1\right)^2}\)

\(\Rightarrow x^2-2x+2014=A\left(x+1\right)^2\)

\(\Leftrightarrow x^2-2x+2014=Ax^2+2Ax+A\)

\(\Leftrightarrow\left(1-A\right)x^2-2\left(A+1\right)x+\left(2014-A\right)=0\)

\(\Delta=4\left(A+1\right)^2-4\left(1-A\right)\left(2014-A\right)\)

\(=8068A-8052\)

Vì A có GTNN nên phương trình có nghiệm

\(\Leftrightarrow8068A-8052\ge0\Leftrightarrow A\ge\frac{2013}{2017}\)

Dấu "=" khi \(x=\frac{2015}{2}\)