Hàm xác định trên R khi và chỉ khi:
\(sin^4x+cos^4x+4sinx.cosx+m-5\ge0;\forall m\)
\(\Leftrightarrow sin^4x+cos^4x+4sinx.cosx-5\ge-m;\forall m\)
\(\Leftrightarrow-m\le\min\limits_{x\in R}f\left(x\right)\)
Với \(f\left(x\right)=sin^4x+cos^4x+4sinx.cosx-5\)
Ta có:
\(f\left(x\right)=\left(sin^2x+cos^2x\right)^2-2sin^2x.cos^2x+4sinx.cosx-5\)
\(=-\dfrac{1}{2}\left(2sinx.cosx\right)^2+2sin2x-4\)
\(=-\dfrac{1}{2}sin^22x+2sin2x-4\)
\(=\dfrac{1}{2}\left(-sin^22x+4sin2x+5\right)-\dfrac{13}{2}\)
\(=\dfrac{1}{2}\left(5-sin2x\right)\left(sin2x+1\right)-\dfrac{13}{2}\ge-\dfrac{13}{2}\) do \(-1\le sin2x\le1\)
\(\Rightarrow\min\limits_{x\in R}f\left(x\right)=-\dfrac{13}{2}\Rightarrow m\ge\dfrac{13}{2}\)
