cho B= √x/(x+ √x+1) a)tính B khi x=28-6√3 b)chứng minh B<1/3 22/07/2021 Bởi Ivy cho B= √x/(x+ √x+1) a)tính B khi x=28-6√3 b)chứng minh B<1/3
Đáp án: $\begin{array}{l}a)B = \dfrac{{\sqrt x }}{{x + \sqrt x + 1}}\left( {dkxd:x \ge 1} \right)\\x = 28 – 6\sqrt 3 \\ \Rightarrow x = 27 – 2.3\sqrt 3 .1 + 1\\ \Rightarrow x = {\left( {3\sqrt 3 – 1} \right)^2}\\ \Rightarrow \sqrt x = 3\sqrt 3 – 1\\B = \dfrac{{3\sqrt 3 – 1}}{{28 – 6\sqrt 3 + 3\sqrt 3 – 1 + 1}}\\ = \dfrac{{3\sqrt 3 – 1}}{{28 – 3\sqrt 3 }} = \dfrac{{81\sqrt 3 – 1}}{{757}}\\b)B < \dfrac{1}{3}\\ \Rightarrow \dfrac{{\sqrt x }}{{x + \sqrt x + 1}} < \dfrac{1}{3}\\ \Rightarrow 3\sqrt x < x + \sqrt x + 1\\ \Rightarrow x – 2\sqrt x + 1 > 0\\ \Rightarrow {\left( {\sqrt x – 1} \right)^2} > 0\\ \Rightarrow \sqrt x \ne 1\\ \Rightarrow x \ne 1\\Vay\,x \ne 1;x \ge 0\end{array}$ Bình luận
Đáp án:
$\begin{array}{l}
a)B = \dfrac{{\sqrt x }}{{x + \sqrt x + 1}}\left( {dkxd:x \ge 1} \right)\\
x = 28 – 6\sqrt 3 \\
\Rightarrow x = 27 – 2.3\sqrt 3 .1 + 1\\
\Rightarrow x = {\left( {3\sqrt 3 – 1} \right)^2}\\
\Rightarrow \sqrt x = 3\sqrt 3 – 1\\
B = \dfrac{{3\sqrt 3 – 1}}{{28 – 6\sqrt 3 + 3\sqrt 3 – 1 + 1}}\\
= \dfrac{{3\sqrt 3 – 1}}{{28 – 3\sqrt 3 }} = \dfrac{{81\sqrt 3 – 1}}{{757}}\\
b)B < \dfrac{1}{3}\\
\Rightarrow \dfrac{{\sqrt x }}{{x + \sqrt x + 1}} < \dfrac{1}{3}\\
\Rightarrow 3\sqrt x < x + \sqrt x + 1\\
\Rightarrow x – 2\sqrt x + 1 > 0\\
\Rightarrow {\left( {\sqrt x – 1} \right)^2} > 0\\
\Rightarrow \sqrt x \ne 1\\
\Rightarrow x \ne 1\\
Vay\,x \ne 1;x \ge 0
\end{array}$