GTNN : (√x+3 – 4√x-1 ) + (√x+15 -8√x-1) (Dấu căn kéo dài đến ” ) “) Question GTNN : (√x+3 – 4√x-1 ) + (√x+15 -8√x-1) (Dấu căn kéo dài đến ” ) “) in progress 0 Môn Toán Kiệt Gia 4 years 2020-10-29T20:23:33+00:00 2020-10-29T20:23:33+00:00 1 Answers 56 views 0
Answers ( )
Đáp án: $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}\ge 2$
Giải thích các bước giải:
ĐKXĐ: $x\ge 1$
Ta có:
$A=\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}$
$\to A=\sqrt{x-1-4\sqrt{x-1}+4}+\sqrt{x-1-8\sqrt{x-1}+16}$
$\to A=\sqrt{(\sqrt{x-1}-2)^2}+\sqrt{(\sqrt{x-1}-4)^2}$
$\to A=|\sqrt{x-1}-2|+|\sqrt{x-1}-4|$
$\to A=|\sqrt{x-1}-2|+|4-\sqrt{x-1}|$
$\to A\ge |\sqrt{x-1}-2+4-\sqrt{x-1}|$
$\to A\ge 2$
Dấu = xảy ra khi $(\sqrt{x-1}-2)(4-\sqrt{x-1})\ge 0$
$\to 2\le \sqrt{x-1}\le 4$
$\to 5\le x\le 17$