The complete set of values of $x$ for which $f(x) = 2 \log_e(x - 2) - x^2 + 4x + 1$ is increasing is:

Find the intervals in which the function $f$ given by $f(x) = x^{3} + \frac{1}{x^{3}}, x \neq 0$ is:
$(i)$ increasing
$(ii)$ decreasing.

For what values of $x$ is the function $f(x) = 2x^3 - 9x^2 + 12x + 29$ decreasing?

Find the intervals in which the function given by $f(x) = \sin 3x, x \in \left[0, \frac{\pi}{2}\right]$ is:
$(a)$ increasing
$(b)$ decreasing.

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 $x \to \infty$.

On the interval $(1, 3)$,the function $f(x) = 3x + \frac{2}{x}$ is