Question

Find in which intervals, f$:$ R$\to$R, $f\left(x\right)=\frac{4\sin x-2x-x\cos x}{2+\cos x}$ is decreasing and intervals in which it is increasing.

Answer 1

Given$f:R\to R,f\left(x\right)=\frac{4sinx-2x-xcosx}{2+cosx}$

$=\frac{4sinx}{2+cosx}-x$

$f^{\prime }\left(x\right)=\frac{(2+cosx)4cosx-4sinx(-sinx)}{(2+cosx{)}^2}-1$

$=\frac{8cosx+4co{s}^{2}x+4si{n}^{2}x}{(2+cosx{)}^2}-1$

$=\frac{8cosx+4}{(2+cosx{)}^2}\le 1$

$\Rightarrow 8cosx+4\le 4+co{s}^{2}x+4cosx$

$\Rightarrow co{s}^{2}x-4cosx⩾0$

$\Rightarrow cosx(cosx-4)⩾0$

$\Rightarrow cosx⩽0\Rightarrow x\epsilon [\frac{\pi }{2}+2n\pi ,2n\pi +\frac{3\pi }{2}]$

$n=....-2,-1,0,\mathrm{12...}nϵ2$

f(x) in creasing for $f^{\prime }\left(x\right)⩾0$

$\Rightarrow \frac{8cosx+4}{(2+cosx{)}^2}⩾1$

$\Rightarrow co{s}^{2}x-4cosx\le 0$

$\Rightarrow 0⩽cosx⩽1$

$\Rightarrow xϵ[2n\pi -\frac{\pi }{2},n\pi +\frac{\pi }{2}]nϵz.$