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

Statement $-1:$ The function $f(x) = x^2(e^x + e^{-x})$ is increasing for all $x > 0.$
Statement $-2:$ The functions $g(x) = x^2e^x$ and $h(x) = x^2e^{-x}$ are increasing for all $x > 0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b).$

Let $f(x) = \begin{cases} x e^{3x}, & x \le 0 \\ 2x^3 + x, & x > 0 \end{cases}$. Find the complete set of values of $x$ for which $f'(x)$ is an increasing function.

The function $f(x) = 2\ln|x| - x|x|$ is increasing on the interval

$f(x) = \int {\left( {2 - \frac{1}{{1 + {x^2}}} - \frac{1}{{\sqrt {1 + {x^2}} }}} \right)} \,dx$. Then $f$ is:

For the function $f(x) = \sin 3x$,where $x \in [0, \frac{\pi}{2}]$,which of the following is true?