$(1)$ Consider the function $f(x,y)$ with formula

$f(x,y)=-x^2+xy-x-y^2+5y+1,(x,y)in{R}^2.$

$($a$)$ Show that its gradient vector equals

gradf$(x,y)=(-2x+y-1,x-2y+5)$

$($b$)$ Show that it has a unique critical point.

$($c$)$ Find the derivatives of order $2.$

$($d$)$ Find the Hessian determinant of $f.$

$($e$)$ Classify the above critical point.