Question

[๖ۣۜIKUN]

Cho biểu thức: 

A=\(\dfrac{\sqrt{x}+1}{x+4\sqrt{x}+4}:\left(\dfrac{x}{x+2\sqrt{x}}+\dfrac{x}{\sqrt{x}+2}\right)\)với x>0

a/ Rút gọn biểu thức A

b/ Tìm tất cả các giá trị x để A≥\(\dfrac{1}{3\sqrt{x}}\)

Answer 1

[Nguyễn Việt Lâm]

\(A=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}+2\right)^2}:\left(\dfrac{x}{\sqrt{x}\left(\sqrt{x}+2\right)}+\dfrac{x}{\sqrt{x}+2}\right)\)

\(=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}+2\right)^2}:\left(\dfrac{\sqrt{x}}{\sqrt{x}+2}+\dfrac{x}{\sqrt{x}+2}\right)\)

\(=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}+2\right)^2}:\left(\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+2}\right)\)

\(=\dfrac{\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)^2}.\dfrac{\left(\sqrt{x}+2\right)}{\sqrt{x}.\left(\sqrt{x}+1\right)}\)

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

\(A\ge\dfrac{1}{3\sqrt{x}}\Leftrightarrow\dfrac{1}{\sqrt{x}\left(\sqrt{x}+2\right)}\ge\dfrac{1}{3\sqrt{x}}\)

\(\Leftrightarrow\dfrac{1}{\sqrt{x}+2}\ge\dfrac{1}{3}\Leftrightarrow\sqrt{x}+2\le3\)

\(\Rightarrow x\le1\)

Kết hợp ĐKXĐ \(\Rightarrow0< x\le1\)