tìm tất cả giá trị thực của m để pt m√(2 + tan²x) = m+ tanx có ít nhất một nghiệm thực 04/10/2021 Bởi Adalyn tìm tất cả giá trị thực của m để pt m√(2 + tan²x) = m+ tanx có ít nhất một nghiệm thực
\[\begin{array}{l} m\sqrt {2 + {{\tan }^2}x} = m + \tan x\,\,\,\left( * \right)\\ DK:\,\,\,\cos x \ne 0\\ Dat\,\,t = \tan x\,\,\left( {t \in R} \right)\\ \Rightarrow pt \Leftrightarrow m\sqrt {2 + {t^2}} = m + t\\ \Leftrightarrow m\left( {\sqrt {2 + {t^2}} – 1} \right) = t\,\,\,\left( * \right)\\ Ta\,\,co\,\,\,\sqrt {2 + {t^2}} – 1 \ne 0\,\,\forall t\\ \Rightarrow \left( * \right) \Leftrightarrow m = \frac{t}{{\sqrt {2 + {t^2}} – 1}}\,\,\,\left( 1 \right)\\ So\,\,nghiem\,\,\,cua\,\,pt\,\,\left( 1 \right)\,\,\,la\,\,\,so\,\,giao\,\,diem\,\,cua\,\,dt\,\,ham\,\,\,so\,\,y = f\left( t \right)\,\,\,va\,\,\,duong\,\,thang\,\,\,y = m.\\ Xet\,\,ham\,\,so:\,\,f\left( t \right) = \frac{t}{{\sqrt {2 + {t^2}} – 1}}\\ \Rightarrow f’\left( t \right) = \frac{{\sqrt {2 + {t^2}} – 1 – \frac{{{t^2}}}{{\sqrt {2 + {t^2}} }}}}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}} = \frac{{2 + {t^2} – \sqrt {2 + {t^2}} – {t^2}}}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}} = \frac{{2\sqrt {2 + {t^2}} }}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}}\\ \Rightarrow f’\left( t \right) = 0 \Leftrightarrow 2 – \sqrt {{t^2} + 2} = 0\\ \Leftrightarrow \sqrt {{t^2} + 2} = 2 \Leftrightarrow {t^2} + 2 = 4 \Leftrightarrow {t^2} = 2 \Leftrightarrow \left[ \begin{array}{l} t = \sqrt 2 \\ t = – \sqrt 2 \end{array} \right.\\ Bang\,\,\,xet\,\,dau:\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \,\,\,\,\,\,\,\,\,\, – \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \\ + \infty \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 2 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \infty \\ \Rightarrow pt\,\,co\,\,\,nghiem\,\,\,thuc\,\,\, \Leftrightarrow – \sqrt 2 \le m \le \sqrt 2 . \end{array}\] Bình luận
\[\begin{array}{l}
m\sqrt {2 + {{\tan }^2}x} = m + \tan x\,\,\,\left( * \right)\\
DK:\,\,\,\cos x \ne 0\\
Dat\,\,t = \tan x\,\,\left( {t \in R} \right)\\
\Rightarrow pt \Leftrightarrow m\sqrt {2 + {t^2}} = m + t\\
\Leftrightarrow m\left( {\sqrt {2 + {t^2}} – 1} \right) = t\,\,\,\left( * \right)\\
Ta\,\,co\,\,\,\sqrt {2 + {t^2}} – 1 \ne 0\,\,\forall t\\
\Rightarrow \left( * \right) \Leftrightarrow m = \frac{t}{{\sqrt {2 + {t^2}} – 1}}\,\,\,\left( 1 \right)\\
So\,\,nghiem\,\,\,cua\,\,pt\,\,\left( 1 \right)\,\,\,la\,\,\,so\,\,giao\,\,diem\,\,cua\,\,dt\,\,ham\,\,\,so\,\,y = f\left( t \right)\,\,\,va\,\,\,duong\,\,thang\,\,\,y = m.\\
Xet\,\,ham\,\,so:\,\,f\left( t \right) = \frac{t}{{\sqrt {2 + {t^2}} – 1}}\\
\Rightarrow f’\left( t \right) = \frac{{\sqrt {2 + {t^2}} – 1 – \frac{{{t^2}}}{{\sqrt {2 + {t^2}} }}}}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}} = \frac{{2 + {t^2} – \sqrt {2 + {t^2}} – {t^2}}}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}} = \frac{{2\sqrt {2 + {t^2}} }}{{{{\left( {\sqrt {2 + {t^2}} – 1} \right)}^2}}}\\
\Rightarrow f’\left( t \right) = 0 \Leftrightarrow 2 – \sqrt {{t^2} + 2} = 0\\
\Leftrightarrow \sqrt {{t^2} + 2} = 2 \Leftrightarrow {t^2} + 2 = 4 \Leftrightarrow {t^2} = 2 \Leftrightarrow \left[ \begin{array}{l}
t = \sqrt 2 \\
t = – \sqrt 2
\end{array} \right.\\
Bang\,\,\,xet\,\,dau:\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \,\,\,\,\,\,\,\,\,\, – \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \\
+ \infty \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 2 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, – \infty \\
\Rightarrow pt\,\,co\,\,\,nghiem\,\,\,thuc\,\,\, \Leftrightarrow – \sqrt 2 \le m \le \sqrt 2 .
\end{array}\]