Để hàm số xác định \(\forall x\in R\Leftrightarrow sin^4x+cos^4x-2msinx.cosx\ge0\) \(\forall x\)
Ta có:
\(sin^4x+cos^4x-2msinx.cosx=\left(sin^2x+cos^2x\right)^2-2\left(sinx.cosx\right)^2-m.sin2x\)
\(=1-2\left(\frac{1}{2}sin2x\right)^2-msin2x=-\frac{1}{2}sin^22x-msin2x+1\)
Xét \(f\left(t\right)=-\frac{1}{2}t^2-mt+1\) với \(t\in\left[-1;1\right]\)
\(f\left(-1\right)=\frac{1}{2}+m\) ; \(f\left(1\right)=\frac{1}{2}-m\)
Để \(f\left(t\right)\ge0\) \(\forall t\in\left[-1;1\right]\Rightarrow\min\limits_{\left[-1;1\right]}f\left(t\right)\ge0\)
\(\Rightarrow\left\{{}\begin{matrix}f\left(-1\right)\ge0\\f\left(1\right)\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}m\ge-\frac{1}{2}\\m\le\frac{1}{2}\end{matrix}\right.\) \(\Rightarrow-\frac{1}{2}\le m\le\frac{1}{2}\)