The values of $a$ for which the function $f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1$ decreases monotonically for all real $x$ are:

The range in which $y = -x^{2} + 6x - 3$ is increasing is

The function $f(x) = 1 - x^3 - x^5$ is decreasing for

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,

If $f(x) = x^5 - 20x^3 + 240x$,then $f(x)$ satisfies which of the following?

Let $f$ be a function defined on $[a, b]$ such that $f^{\prime}(x) > 0$ for all $x \in (a, b)$. Prove that $f$ is an increasing function on $(a, b)$.