The function \(f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}\) is increasing, if

\(f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x} \)
\( f^{\prime}(x)=\frac{\{[(2 \sin x+3 \cos x)(\lambda \cos x-6 \sin x]-[\lambda \sin x+6 \cos x)(2 \cos x-3 \sin x)]\}}{(2 \sin x+3 \cos x)^2}\)
When \(\mathrm{f}^{\prime}(\mathrm{x}) \geq 0\), we get
\([(2 \lambda \sin x \cos x+3 \lambda \cos ^2 x-12 \sin ^2 x-18 \sin x\) \(\cos x)]\)
\( -(2 \lambda \sin \mathrm{x} \cos \mathrm{x}+12 \cos ^2 \mathrm{x}-3 \lambda \sin ^2 \mathrm{x}-18 \operatorname{sun} \mathrm{x}\) \(\cos \mathrm{x})]\) \(\geq 0 \)
\( \therefore 3 \lambda\left(\sin ^2 \mathrm{x}+\cos ^2 \mathrm{x}\right)-12\left(\sin ^2 \mathrm{x}+\cos ^2 \mathrm{x}\right) \geq 0 \)
\( \therefore 3 \lambda-12 \geq 0 \Rightarrow \lambda \geq 4\)