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

Let $$f(x) = x + {1 \over x}$$, where x$$\in$$ (0, 1). Then which one of the following is correct?

Let the slope of the curve y = cos^$$-$$1 (sin x) be tan$$\theta$$. Then, the value of $$\theta$$ in the interval (0, $$\pi$$) is

Which one of the following is correct in respect of the function $$f(x) = x\sin x + \cos x + {1 \over 2}{\cos ^2}x$$ ?