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

The function $y = 2x^3 - 9x^2 + 12x - 6$ is monotonically decreasing when

For every value of $x$,the function $f(x) = \frac{1}{5^x}$ is:

Prove that the function given by $f(x) = \cos x$ is increasing in $(\pi, 2\pi)$.

Let $f(x) = \sin x$ and $g(x) = x$.
Statement $1$: $f(x) \le g(x)$ for $x$ in $(0, \infty)$.
Statement $2$: $f(x) \le 1$ for $x$ in $(0, \infty)$ but $g(x) \to \infty$ as $x \to \infty$.

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right)$,where $x \in\left[0, \frac{\pi}{2}\right]$,consider the following two statements:
$(I)$ $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.
$(II)$ $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
Which of the following is correct?