Monotonically Decreasing Functions

Q. Let $f:R\to R$ be a continuous decreasing function. A point $x_0\in R$ is said to be a fixed point of $f$ if $f\left(x_0\right)=x_0.$
The number of fixed points of $f\circ f\circ f$ equals

1. $0$
2. $1$
3. $2$
4. infinitely many

Q. Let f(x) be a function such that $f\text{'}\left(x\right)={\log }_{1/3}\left({\log }_3\left(\sin x+a\right)\right]$. If f(x) is decreasing for all real values of x, then .

1. $a\in \left(1,4\right)$
2. $a\in \left(4,\infty \right)$
3. $a\in \left(2,3\right)$
4. $a\in \left(2,\infty \right)$

Q. The set of all real values of $a$ for which the function $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases monotonically throughout for all real $x,$ is

1. $a<-2$
2. $a>-2$
3. $-3\le a<0$
4. $a\le -3$

Q. Function $g\left(x\right)=9x-4\tan x,x\in (0,\frac{\pi }{2})$ will have

1. No local extremum
2. Local maximum at $x=\frac{1}{2}$
3. Local maximum at ${\cos }^{-1}(-\frac{2}{3})$
4. Local maximum at ${\cos }^{-1}\left(\frac{2}{3}\right)$

Q. Let $f\left(x\right)$ be a non-negative function. If $f^{\prime }\left(x\right)\cos x\le f\left(x\right)\sin x,\forall x\ge 0,$ then value of $f\left(\frac{5\pi }{3}\right)$ is

1. $1$
2. $\frac{1}{2}$
3. $0$
4. $\frac{\sqrt{3}}{2}$