Answer the following for the given function $f$ and region $D.$

$($a$)$ Find and classify all critical points as a local minimum, local maximum, or saddle point.

$($b$)$ Find the maximum and minimum of $f$ over $D.$

$($i$)f(x,y)=x^3-3x{y}^2-3x+y^2$;$D=\{(x,y)$:$x0,y0,x+y4\}$

$($ii$)f(x,y)=4x^2-y^2-4x+2y$;$D=\{(x,y)$:$-2x2,-2y2\}$

$($iii$)f(x,y)=x^2+y^2-2x+4y$;$D=\{(x,y)$:$x^2+y^{2}4\}$

$($iv$)f(x,y)=x^3-3xy+y^3$;$D=\{(x,y)$:$x^2+y^{2}9\}$

$($v$)f(x,y)=x^2-y^2+4x-2y$;$D=\{(x,y)$:$x^2+4y^{2}16\}$

$($vi$)f(x,y)=x^2+y^2+2x-3y$;$D=\{(x,y)$:$x^2+y^2-4x0\}$