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}\]