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Mathematics · 2023 · 2 marks
CBSE 2023 · Region 1 · Set 1 · Q25
Show that the function f(x)=(16 sin x)/(4+ cos x)-x, is strictly decreasing in ((π)/2, π).
Show that the function $\displaystyle f(x)=\frac{16 \sin x}{4+\cos x}-x$, is strictly decreasing in $\displaystyle \left(\frac{\pi}{2}, \pi\right)$.
Marking-scheme solution
$$\begin{aligned}
& f^{\prime}(x)=\frac{16[4+\cos x] \cos x+16 \sin ^{2} x}{(4+\cos x)^{2}}-1 \\
& =\frac{\cos x(56-\cos x)}{(4+\cos x)^{2}} \\
& \text { in }\left(\frac{\pi}{2}, \pi\right), \cos x<0 \Rightarrow f^{\prime}(x)<0 \\
& \therefore f(x) \text { in strictly decreasing in }\left(\frac{\pi}{2}, \pi\right)
\end{aligned}
$$
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