Bonus Refer to $Q(2b)$ for the symbols and the results derived there.

$(5+5)+5=15$

$($a$)$ Comment on the signs of

$\frac{\text{d}\text{e}\text{l}}{deln}{P}_{\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}}(n,p)$

and

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{p}}{P}_{\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}}(n,p)$

$($b$)$ Does it give you insight to choose $n$ for a given $p$ and desired $P_{\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}}(n,p)?$