Let $R$ be the set of all real numbers and $f(x)=\sin ^{10} x\left(\cos ^8 x+\cos ^4 x+\cos ^2 x+1\right)$ $x \in R$. Let  $S=\{\lambda \in R$ there exists a point $c \in(0,2 \pi)$ with $\left.f^{\prime}(c)=\lambda f(c)\right\}$ Then,

Domain of the function $f(x) = \sqrt {2 – {{\sec }^{ – 1}}x} $ is 

Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x)=\frac{\{x\}}{1+[x]^2}$, where $[x]$ is the greatest integer less than or equal to $x$, and $\left\{x{\}}=x-[x]\right.$. Which of the following statements are true?

$I.$ The range of $f$ is a closed interval.

$II.$ $f$ is continuous on $R$.

$III.$ $f$ is one-one on $R$

The period of the function $f(x) = \log \cos 2x + \sin 4x$ is :-

If $f({x_1}) – f({x_2}) = f\left( {\frac{{{x_1} – {x_2}}}{{1 – {x_1}{x_2}}}} \right)$ for ${x_1},{x_2} \in [ – 1,\,1]$, then $f(x)$ is