Let $R^* = R - \left\{ (2k - 1) \frac{\pi}{2} \mid k \in I \right\}$. The function $f: R^* \rightarrow R$ is defined as $f(x) = \tan x - x$, then $f(x)$ is

The function $\sin x - \cos x$ is increasing in the interval

Consider the following statements $S$ and $R$:
$S$: Both $\sin x$ and $\cos x$ are decreasing functions in $\left( \frac{\pi}{2}, \pi \right)$.
$R$: If a differentiable function decreases in $(a, b)$,then its derivative also decreases in $(a, b)$.
Which of the following is true?

If $f''(x) < 0$ for all $x \in (0, 2)$,then the function $H(x) = f(1 - x) + 2f(x/2)$ is:

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

If $f(x)=2x^3-15x^2-144x-7$,then $f(x)$ is strictly decreasing in