Question

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

a, rút gọn P

b, tìm GTNN, GTLN của P

c. tính P tại x= 3-2\(\sqrt{2}\)

d, tìm x để P>0

e. so sánh p vs \(-2\sqrt{x}\)

Answer 1

ĐKXĐ:\(x>0,x\ne1\)

\(\Leftrightarrow P=\left(\dfrac{\left(\sqrt{x}-1\right)^2-\left(\sqrt{x}+1\right)^2}{x-1}\right)\cdot\left(\dfrac{1-x}{2\sqrt{x}}\right)^2\)

\(\Leftrightarrow P=\left(\dfrac{x-2\sqrt{x}+1-x-2\sqrt{x}-1}{x-1}\right)\cdot\dfrac{\left(1-x\right)^2}{4x}\)

\(\Leftrightarrow P=\dfrac{-4\sqrt{x}}{x-1}\cdot\dfrac{\left(x-1\right)^2}{4x}\)

\(\Leftrightarrow P=\dfrac{1-x}{\sqrt{x}}\)