Question

[Trần Bảo Hân]

a) Tìm giá trị nhỏ nhất của biểu thức: x - \(4\sqrt{x-2009}-2005\)

b) Tìm x, y, z biết: \(\frac{\sqrt{x-2009}-1}{x-2009}+\frac{\sqrt{y-2010}-1}{y-2010}+\frac{\sqrt{z-2011}-1}{z-2011}=\frac{3}{4}\)

Answer 1

[Nguyễn Việt Lâm]

ĐKXĐ: ...

a/ \(A=x-2009-4\sqrt{x-2009}+4=\left(\sqrt{x-2009}-2\right)^2\ge0\)

\(A_{min}=0\) khi \(\sqrt{x-2009}-2=0\Rightarrow x=2013\)

b/ \(\frac{1}{4}-\frac{\sqrt{x-2009}-1}{x-2009}+\frac{1}{4}-\frac{\sqrt{y-2010}-1}{y-2010}+\frac{1}{4}-\frac{\sqrt{z-2011}-1}{z-2011}=0\)

\(\Leftrightarrow\frac{x-2009-4\sqrt{x-2009}+4}{4\left(x-2009\right)}+\frac{y-2010-4\sqrt{y-2010}+4}{4\left(y-2010\right)}+\frac{z-2011-4\sqrt{z-2011}+4}{4\left(z-2011\right)}=0\)

\(\Leftrightarrow\frac{\left(\sqrt{x-2009}-2\right)^2}{4\left(x-2009\right)}+\frac{\left(\sqrt{y-2010}-2\right)^2}{4\left(y-2010\right)}+\frac{\left(\sqrt{z-2011}-2\right)^2}{4\left(z-2011\right)}=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x-2009}-2=0\\\sqrt{y-2010}-2=0\\\sqrt{z-2011}-2=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=2013\\y=2014\\z=2015\end{matrix}\right.\)