The function $f(x) =$ ${x^{\frac{1}{{\ln \,x}}}}$

Which pair $(s)$ of function $(s)$ is/are equal ?

where $\{x\}$ and $[x]$ denotes the fractional part $\&$ integral part functions.

Range of the function

$f(x) = \sqrt {\left| {{{\sin }^{ – 1}}\left| {\sin x} \right|} \right| – {{\cos }^{ – 1}}\left| {\cos x} \right|} $ is

Let $2{\sin ^2}x + 3\sin x – 2 > 0$ and ${x^2} – x – 2 < 0$ ($x$ is measured in radians). Then $x$  lies in the interval

If non-zero real numbers $b$ and $c$ are such that $min \,f\left( x \right) > \max \,g\left( x \right)$, where $f\left( x \right) = {x^2} + 2bx + 2{c^2}$  and $g\left( x \right) = {-x^2} – 2cx + {b^2}$$\left( {x \in R} \right)$; then $\left| {\frac{c}{b}} \right|$ lies in the interval