Đặt \(\sqrt{3+x}+\sqrt{6-x}=t\Rightarrow3\le t\le3\sqrt{2}\)
\(t^2=9+2\sqrt{\left(3+x\right)\left(6-x\right)}\Rightarrow\sqrt{\left(3+x\right)\left(6-x\right)}=\frac{t^2-9}{2}\)
BPT trở thành:
\(t-\frac{t^2-9}{2}\le m\) ; \(\forall t\in\left[3;3\sqrt{2}\right]\)
\(\Leftrightarrow f\left(t\right)=-\frac{1}{2}t^2+t+\frac{9}{2}\le m\) ; \(\forall t\in\left[3;3\sqrt{2}\right]\)
\(\Leftrightarrow m\ge\max\limits_{\left[3;3\sqrt{2}\right]}f\left(t\right)\)
\(-\frac{b}{2a}=1\notin\left[3;3\sqrt{2}\right]\) ; \(f\left(3\right)=3\) ; \(f\left(3\sqrt{2}\right)=\frac{6\sqrt{2}-9}{2}< 3\)
\(\Rightarrow\max\limits_{\left[3;3\sqrt{2}\right]}f\left(t\right)=3\Rightarrow m\ge3\)
