The complete set of values of $m$ for which the function $f(x) = e^{\sin x} + 2m\sin x + 1$ is increasing for all $x \in \left( 0, \frac{\pi}{2} \right)$ is:

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x > 0 \\ 3xe^x, & x \leq 0 \end{cases}$. Then $f$ is an increasing function in the interval:

The function $f(x) = 2x^3 - 9x^2 + 12x + 2$ is decreasing in

If $f(x) = x^2 + kx + 1$ is an increasing function on the interval $[1, 2]$,what is the minimum value of $k$?

Consider the following statements $S$ and $R$:
$S: \sin x$ and $\cos x$ are both decreasing functions in the interval $\left( \frac{\pi}{2}, \pi \right)$.
$R:$ If a function $f$ is decreasing in an interval $(a, b)$,then its derivative $f'(x)$ is also decreasing in the interval $(a, b)$.
Which of the following is true?

Which of the following is true about $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$?