$[$Maximum points: $12]$

Let $f(x)=x^2-4x+3.$

The diagram below shows the area bound by the graph of $y=f(x),$ the normal to the graph at the $y-$intercept, and the $x-$axis.

$($a$)$ Find $f^{\text{'}}(x).$

$($b$)$ Find

$($i$)$ the equation of the normal line in the form $y=mx+c$ where $m,$cinR

$($ii$)$ the coordinates of the $x-$intercept of the normal line

$($c$)$ Solve $f(x)=0.$

$($d$)$ Find the area of the shaded region.