Hàm số xác định trên R khi và chỉ khi:
\(sin^2x+\left(2m-3\right)cosx+3m-2>0;\forall x\in R\)
\(\Leftrightarrow-cos^2x+\left(2m-3\right)cosx+3m-1>0\)
\(\Leftrightarrow t^2-\left(2m-3\right)t-3m+1< 0;\forall t\in\left[-1;1\right]\)
\(\Leftrightarrow t^2+3t+1< m\left(2t+3\right)\)
\(\Leftrightarrow\dfrac{t^2+3t+1}{2t+3}< m\) (do \(2t+3>0;\forall t\in\left[-1;1\right]\))
\(\Leftrightarrow m>\max\limits_{\left[-1;1\right]}\dfrac{t^2+3t+1}{2t+3}\)
Ta có: \(\dfrac{t^2+3t+1}{2t+3}=\dfrac{t^2+t-2+2t+3}{2t+3}=\dfrac{\left(t-1\right)\left(t+2\right)}{2t+3}+1\)
Do \(-1\le t\le1\Rightarrow\dfrac{\left(t-1\right)\left(t+2\right)}{2t+3}\le0\)
\(\Rightarrow\max\limits_{\left[-1;1\right]}\dfrac{t^2+3t+1}{2t+3}=1\)
\(\Rightarrow m>1\)
