Question

Cho \(A=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\) (x≥1)

a, Rút gọn A

b, Tìm MinA

Answer 1

a) \(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)

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

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

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

b) \(\left|\sqrt{x-1}+1\right|+\left|\sqrt{x-1}-1\right|\)

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

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

Dấu "=" xảy ra \(\Leftrightarrow1\le x\le2\)