If $\mathrm{f}(x)=\frac{\mathrm{a} \sin x+b \cos x}{\mathrm{c} \sin x+\mathrm{d} \cos x}$ is decreasing for…
If $\mathrm{f}(x)=\frac{\mathrm{a} \sin x+b \cos x}{\mathrm{c} \sin x+\mathrm{d} \cos x}$ is decreasing for all $x$, then
- $\mathrm{ad}-\mathrm{bc}\gt0$
- $\mathrm{ad}-\mathrm{bc} \lt 0$
- $\mathrm{ab}-\mathrm{cd}\gt0$
- ab $-\mathrm{cd} \lt 0$
Solution
$\begin{aligned} & \mathrm{f}(x)=\frac{\mathrm{a} \sin x+\mathrm{b} \cos x}{\mathrm{c} \sin x+\mathrm{d} \cos x} \\ & \mathrm{f}(x) \text { will be decreasing, if } \mathrm{f}^{\prime}(x) \lt 0 \\ \therefore \quad & \frac{1}{(\operatorname{csin} x+\mathrm{d} \cos x)^2}[(\mathrm{csin} x+\mathrm{d} \cos x)(\mathrm{a} \cos x-b \sin x)\end{aligned}$
$-(a \sin x+b \cos x)(\cos x-d \sin x)] \lt 0$
$\begin{aligned} & \Rightarrow \operatorname{acsin} x \cos x-\mathrm{bc} \sin ^2 x+\mathrm{ad} \cos ^2 x \\ & \quad-\mathrm{bd} \sin x \cos x-\mathrm{acsin} x \cos x+\mathrm{ad} \sin ^2 x\end{aligned}$
$-\mathrm{bc} \cos ^2 x+\mathrm{bdsin} x \cos x \lt 0$
$\begin{aligned} & \Rightarrow \mathrm{ad}\left(\sin ^2 x+\cos ^2 x\right)-\mathrm{bc}\left(\sin ^2 x+\cos ^2 x\right) \lt 0 \\ & \Rightarrow \mathrm{ad}-\mathrm{bc} \lt 0\end{aligned}$
Asked in: MHT CET 2024 (16 May Shift 2)
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