If $f(x) = x^{3/2}(3x - 10)$,$x \geq 0$,then in which interval is $f(x)$ an increasing function?

If $f(x) = \frac{a \sin x + b \cos x}{c \sin x + d \cos x}$ is decreasing for all $x$,then

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

The range of values of $x$ for which $f(x)=x^3+6x^2-36x+7$ is increasing is:

Let the function $f(x) = \frac{x}{3} + \frac{3}{x} + 3$,$x \neq 0$ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $\sum_{i=1}^5 \alpha_i^2$ is equal to :-

The interval for which the given function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is decreasing,is