In the interval $\left( -\frac{\pi}{3}, \frac{\pi}{3} \right)$,the function $f(x) = -\frac{x}{2} + \sin x$ is:

Let $g(x) = 2f(\frac{x}{2}) + f(2 - x)$ and $f''(x) < 0$ for all $x \in (0, 2)$. Then $g(x)$ increases in

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

The polynomial equation $x^3-3ax^2+(27a^2+9)x+2016=0$ has

If $R$ is the least value of $a$ such that the function $f(x) = x^{2} + ax + 1$ is increasing on $[1, 2]$ and $S$ is the greatest value of $a$ such that the function $f(x) = x^{2} + ax + 1$ is decreasing on $[1, 2]$,then the value of $|R - S|$ is ..... .

In which interval is the given function $f(x) = 2x^3 - 15x^2 + 36x + 1$ monotonically decreasing?