The function $f(x) = 2x^3 - 9x^2 + 12x + 29$ is monotonically decreasing when:

Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is
$(a)$ increasing
$(b)$ decreasing

What is the length of the longest interval in which the function $f(x) = 3\sin x - 4\sin^3 x$ is an increasing function?

$y=x^3-a x^2+48 x+7$ is an increasing function for all real values of $x$,then $a$ lies in the interval

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

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.