tìm m để hàm số y= $\sqrt[]{2sin^2x+4sinxcosx-(3+2m)cos^2x+2}$ xác định với mọi x

tìm m để hàm số
y= $\sqrt[]{2sin^2x+4sinxcosx-(3+2m)cos^2x+2}$ xác định với mọi x

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  1. \[\begin{array}{l}
    y = \sqrt {2{{\sin }^2}x + 4\sin x\cos x – \left( {3 + 2m} \right){{\cos }^2}x + 2} \\
    Ham\,\,so\,\,xd\,\,voi\,\,moi\,\,x\\
    \Leftrightarrow f\left( x \right) = 2{\sin ^2}x + 4\sin x\cos x – \left( {3 + 2m} \right){\cos ^2}x + 2 \ge 0\,\,\forall x\,\\
    Voi\,\,\cos x = 0\\
    \Rightarrow f\left( x \right) = 2{\sin ^2}x + 2 > 0\,\,\\
    \Rightarrow \cos x = 0\,\,khong\,\,la\,\,nghiem\,\,cua\,pt\,\,f\left( x \right) = 0.\\
    Chia\,\,ca\,\,2\,\,ve\,\,cua\,\,\left( * \right)\,cho\,\,{\cos ^2}x\,\,ta\,\,duoc:\\
    2{\tan ^2}x + 4\tan x – 3 – 2m + \frac{2}{{{{\cos }^2}x}} \ge 0\,\,\forall x\\
    \Leftrightarrow \,\,2{\tan ^2}x + 4\tan x – 3 – 2m + 2{\tan ^2}t + 2 \ge 0\,\,\forall t\\
    \Leftrightarrow 4{\tan ^2}x + 4\tan x – 1 – 2m \ge 0\,\,\,\,\,\forall t\,\,\left( 1 \right)\\
    Dat\,\,\tan \,x = t\\
    \Rightarrow \left( 1 \right) \Leftrightarrow 4{t^2} + 4y – 2m – 1 \ge 0\,\,\forall t\\
    \Leftrightarrow \Delta ‘ \le 0 \Leftrightarrow 4 + 4\left( {2m + 1} \right) \le 0\\
    \Leftrightarrow 1 + 2m + 1 \le 0\\
    \Leftrightarrow 2m \le – 2\\
    \Leftrightarrow m \le – 1.
    \end{array}\]

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