Monotonicity Questions

Consider the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$, where $[\cdot ]$ represents the greatest integer function. The function $f(x)$ is decreasing in the interval:

The value of $a$ so that $f(x)=\sin x-\cos x-ax+b$ decreases for all real values of $x$ is given by:

If $f(x)$ is an increasing function and $g(x)$ is a decreasing function, then which of the following is correct for the real values $p$ and $q$? Given that for these values of $x$, both functions are defined.

What is the interval over which the function $f(x)=6x-x^2,x>0$ is increasing?