Let $A=[a, \infty)$ denotes the domain, then $f:(a, \infty) \rightarrow B$, which is defined by $f(x)=2 x^3-3 x^2+6$ will have an inverse for the smallest real value of ' $a$ ' if

Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in(0,90$ ); (where [.] denotes the greatest integer function) is

If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f\left(e^x\right)+f(\ln |x|)$ is

If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a>0$, it satisfies $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2} \forall x$, then $f(x)$ is periodic with period