How many positive real numbers $x$ satisfy the equation $x^3-3|x|+2=0$?

Statement $-1$: Any function $f(x)$ is an even function if $f(-x) = f(x)$ for all $x$ in its domain.
Statement $-2$: The function $f(x) = \frac{1}{\sqrt{1 - x^2}} + \left[ \frac{x^2 + x + 1}{4} \right]$,where $[.]$ denotes the greatest integer function,is an even function.

Let $f(x) = 2x^{2} - x - 1$ and $S = \{n \in \mathbb{Z} : |f(n)| \leq 800\}$. Then the value of $\sum_{n \in S} f(n)$ is:

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$,and let $\{x\} = x - [x]$. The number of solutions $x$ to the equation $[x]\{x\} = 5$ with $0 \leq x \leq 2015$ is

Let the function $g : (-\infty, \infty) \to \left( - \frac{\pi}{2}, \frac{\pi}{2} \right)$ be given by $g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2}$. Then,$g$ is -

If $f(x) = \cos([\pi^2]x) + \cos([- \pi^2]x)$,then which of the following is true?