ANSWER $1$ AND $2$

Let $f(x)=\frac{x^2}{x^2+2}.$

$($a$)$ Compute $f^{\text{'}}(x)$ and determine where $f(x)$ is increasing$/$decreasing$.$

$($b$)$ Find possible inflection points.

$($c$)$ Determine the limit of $f(x)$ as $x+-,$ and sketch the graph of $f(x).$

A firm's production function is $Q(L)=12L^2-\frac{1}{20}{L}^3,$ where $L$ denotes the number of workers, with Lin$[0,200].$

$($a$)$ What size of the work force maximizes output $Q(L)?$

$($b$)$ Let $L^{**}$ denote the size of the work force that maximizes output per worker, $Q\frac{L}{L}.$ Find $L^{**}$ and note that $Q^{\text{'}}(L^{**})=Q\frac{L^{**}}{L^{**}}.$ Is this a coincidence?