If $f(x) = kx^3 - 3x^2 - 12x + 8$ is strictly decreasing for all $x \in R$, then:

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?

The real valued function $f(x) = \frac{x^2}{2} - \log(x^2 + x + 1)$ is

Let $f(x) = \int e^x (x - 1)(x - 2) dx$. Then $f$ decreases in the interval -

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

Let $\phi(x) = f(x) + f(2a - x)$, $x \in [0, 2a]$ and $f^{\prime \prime}(x) > 0$ for all $x \in [0, a]$. Then $\phi(x)$ is