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) strictly increasing
(ii) strictly decreasing

Answer 1

$f\left(x\right)=\frac{4\sin x-2x-x\cos x}{2+\cos x}$

Differentiate w.r.t x, we get

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

or

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

as we know that

$(4-\cos x)>0$

(i)

So, for decreasing function

$\cos x<0$ or $2n\pi +\frac{\pi }{2}$

(ii)

for increasing function

$2n\pi -\frac{\pi }{2}$