Question

Find the intervals in which the function $f$ given by
$f\left(x\right)=\frac{4\sin x-2x-x\cos x}{2+\cos x}$
is
(i) increasing (ii) decreasing.

Answer 1

a function f(x) is increasing if f'(x)>0 and decreasing if f'(x)<0
$f\left(x\right)=\frac{4\sin x-2x-x\cos x}{2+\cos x}{f}^{\prime }\left(x\right)=\frac{x\sin x+3\cos x-2}{2+\cos x}+\frac{\left(\sin x\right)(4\sin x-2x-x\cos x)}{(2+\cos x{)}^2}=\frac{4{\sin }^{2}x+3{\cos }^{2}x+4\cos x-4}{(\cos x+2{)}^2}\frac{4(1-{\cos }^{2}x)+3{\cos }^{2}x+4\cos x-4}{(\cos x+2{)}^2}=\frac{4\cos x-{\cos }^{2}x}{(\cos x+2{)}^2}=\frac{\cos x(4-\cos x)}{(2+\cos x{)}^2}$
$f^{\prime }\left(x\right)=\frac{\cos x(4-\cos x)}{(2+\cos x{)}^2}$
Sign $f^{\prime }\left(x\right)$ only depends on cosx because $(4-\cos x)$ and $(2+\cos x{)}^2$ are always greater than $0$.
$f^{\prime }\left(x\right)$ when $\cos x>0$ and $f^{\prime }\left(x\right)<0$ when $\cos x<0$