Bài 1:

Thay x=9 vào biểu thức \(A=\frac{2\sqrt{x}+1}{\sqrt{x}+2}\), ta được:

\(\frac{2\cdot\sqrt{9}+1}{\sqrt{9}+2}=\frac{2\cdot3+1}{3+2}=\frac{7}{5}\)

Vậy: \(\frac{7}{5}\) là giá trị của biểu thức \(A=\frac{2\sqrt{x}+1}{\sqrt{x}+2}\) tại x=9

Bài 2:

a) Ta có: \(B=\left(\frac{x+14\sqrt{x}-5}{x-25}+\frac{\sqrt{x}}{\sqrt{x}+5}\right):\frac{\sqrt{x}+2}{\sqrt{x}-5}\)

\(=\left(\frac{x+14\sqrt{x}-5}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}+\frac{\sqrt{x}\left(\sqrt{x}-5\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\right)\cdot\frac{\sqrt{x}-5}{\sqrt{x}+2}\)

\(=\frac{2x+9\sqrt{x}-5}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}\cdot\frac{\sqrt{x}-5}{\sqrt{x}+2}\)

\(=\frac{2x+10\sqrt{x}-\sqrt{x}-5}{\sqrt{x}+5}\cdot\frac{1}{\sqrt{x}+2}\)

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

1, a, ĐKXĐ: x > 0

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

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

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

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

\(\Rightarrow P=x-\sqrt{x}\)

b, Thay x=100 vào biểu thức P, ta có:

P= 100 - \(\sqrt{100}\)

\(\Rightarrow P=100-10=90\)

Vậy với x=100 thì P=90

c, Ta có: P= \(x-\sqrt{x}=\left(\sqrt{x}-\frac{1}{2}\right)^2-\frac{1}{4}\ge-\frac{1}{4}\)

Dấu "=" xảy ra khi...

2, a, ĐKXĐ: x \(\ge\) 0, x \(\ne\) 1

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

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

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

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

b, Để \(\frac{1}{A}\)là số tự nhiên (x \(\ge0\), \(x\ne1\))

\(\Rightarrow x-1=1\)

\(\Rightarrow x=2\)

Vậy x=2 thì \(\frac{1}{A}\) là số tự nhiên.

\(a,A=\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

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

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

\(\Rightarrow\frac{3x+3}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(b,Tacó:P=\frac{A}{B}=\frac{3x+3}{2\left(x+\sqrt{x}+1\right)}\)

\(\Rightarrow P=\frac{3}{2}.\frac{x+1}{x+\sqrt{x}+1}\)

\(\Rightarrow P=\frac{3}{2}.\frac{x+1}{x+1+\sqrt{x}}\)

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

\(\Rightarrow P\le\frac{3}{2}.\left(1-0\right)\)

\(\Rightarrow P\le\frac{3}{2}\)

\(\Rightarrow Max_P=\frac{3}{2}\)

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

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

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

Do \(x+\sqrt{x}+1=x+\sqrt{x}+\frac{1}{4}+\frac{3}{4}=\left(\sqrt{x}+\frac{1}{2}\right)^2+\frac{3}{4}>0\)

\(\Rightarrow P=\frac{2}{x+\sqrt{x}+1}>0\)

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

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

\(M=-\frac{1}{2}\Leftrightarrow\frac{x}{\sqrt{x}-1}=-\frac{1}{2}\Leftrightarrow2x=1-\sqrt{x}\)

\(\Leftrightarrow2x+\sqrt{x}-1=0\Leftrightarrow\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)=0\)

\(\Leftrightarrow2\sqrt{x}-1=0\Rightarrow\sqrt{x}=\frac{1}{2}\Rightarrow x=\frac{1}{4}\)

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

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

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

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

b/ \(\frac{x}{\sqrt{x}+1}=\frac{-1}{2}\Leftrightarrow2x=-\sqrt{x}-1\Leftrightarrow2x+\sqrt{x}+1=0\) (vô n)

a/ \(P=\frac{2}{\sqrt{x}-2}:\left(\frac{\sqrt{x}}{x-4}+\frac{1}{\sqrt{x}-2}\right)\left(ĐKXĐ:x\ge0,x\ne4\right)\)

\(\Leftrightarrow P=\frac{2}{\sqrt{x}-2}:\left(\frac{\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right)\)

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

b/ \(P=\frac{\sqrt{x}+1+1}{\sqrt{x}+1}=1+\frac{1}{\sqrt{x}+1}\)

$P$ đạt giá trị lớn nhất \(\Leftrightarrow\left(\sqrt{x}+1\right)\) đạt GTNN

Vì \(\sqrt{x}\ge0\Leftrightarrow\sqrt{x}+1\ge1\) đạt giá trị nhỏ nhất là $1$ tại \(x=0\)

Vậy \(MaxP=2\Leftrightarrow x=0\)

KL: ...................