Question

Let ${}^{\prime }{n}^{\prime }$ be the number of elements in the domain set of the function $f\left(x\right)=\left|ln\sqrt{{}^{x^2+4x}{C}_{2x^2+3}}\right|$,
and ${}^{\prime }{Y}^{\prime }$ be the global maximum value of $f\left(x\right)$, then $[n+[Y\left]\right]$ is ....(where $\left[\right]=$ Greatest Integer Function).

Answer 1

Clearly, $x^2+4x\ge 0$
$2x^2+3\ge 0\Rightarrow x^2+4x\ge 2x^2+3$
and x is an integer,
$\therefore x\in \{1,2,3\}\therefore n=3$
Now, maximum value ${}^{x^2+4x}{\text{C}}_{2x^2+3}=12$
$\therefore Y=\left|\ln 12\right|$
$\therefore \left[Y\right]=2(\ln c^2<\ln 12<\ln c^3)$
$\therefore [n+[Y\left]\right]=[3+2]=5$