The interval in which the function $f(x) = x^{3} - 6x^{2} + 9x + 10$ 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.

In which of the following intervals is the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ strictly decreasing?

Let $g(x) = 2f(\frac{x}{2}) + f(2 - x)$ and $f''(x) < 0$ for all $x \in (0, 2)$. Then $g(x)$ increases in

Let $f : R \rightarrow R$ be a twice differentiable function such that $f^{\prime\prime}(x) > 0$ for all $x \in R$ and $f^{\prime}(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^{2}x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$. Consider the following two statements:
$(I)$ $g$ is increasing in $(0, \frac{\pi}{4})$
$(II)$ $g$ is decreasing in $(\frac{\pi}{4}, \frac{\pi}{2})$
Then,

The function $f(x) = \cos |x| - 2ax + b$ increases on the entire real line. What is the range of $a$?