Let $g(t)=\frac{3t^3}{t^2-3}$ with $g^{\text{'}}(t)=\frac{3t^2(t^2-9)}{(t^2-3{)}^2}$ and $g^{\text{'}\text{'}}(t)=\frac{18t(t^{2}9)}{(t^2-3{)}^3}.$

$($a$)$ Determine the domain of $g.$

$($b$)$ Determine the intervals of increase and decrease.

$($c$)$ Determine the intervals of concave up and concave down.

$($d$)$ Use your work above to answer the following:

The graph of $g$ is concave up and increasing on the interval

The graph of $g$ is concave up and decreasing on the interval

The graph of $g$ is concave down and increasing on the interval

The graph of $g$ is concave down and decreasing on the interval

The graph of $g$ has inflection point$($s$)$ at

$1$

Use calculus to determine the largest possible area of a rectangle with perimeter $48$ cm $.$